x5x           x5x                 
                        F5o                        
                                                   
            o5F                     o5F            
     x5x                                   x5x     
                        F5x                        
                 x5F           x5F                 
                                                   
     F5o                                   F5o     
            F5x                     F5x            
                                                   
x5x                                             x5x
                                                   
            x5F                     x5F            
     o5F                                   o5F     
                                                   
                 F5x           F5x                 
                        x5F                        
     x5x                                   x5x     
            F5o                     F5o            
                                                   
                        o5F                        
                 x5x           x5x                 

The relation to ex runs as follows: ex in pentagonal duoprismatic subsymmetry can be given as xfooxo5xofxoo oxofox5ooxofx&#zx (F = ff = f+x, V = uf = f+f). That will be transformed into (-x)fooxo5Fofxoo oxofox5ooxofx&#zx. Then into (-x)fofxo5Fof(-x)oo oxofox5ooxofx&#zx. And finally into (-x)fof(-x)o5Fof(-x)fo oxofox5ooxofx&#zx. Then a Stott expansion wrt. the first and second nodes produces oFxFox5AxFoFx oxofox5ooxofx&#zx. This polychoron then is a pentadiminishing thereof, in fact thereby just omitting the first layer of that tegum sum.

The decagonal periodic -dip-{10}-dip- ring clearly is obvious here. Then there is an orthogonal, alternating -pip-{5}-pap-gyro{5}-gyropip-gyro{5}-pap-{5}-pip- ring. And, what is quite notable, the pentagons of the bilbiroes just connect two of those each. In fact the latter ones get squeezed between those 2 orthogonal rings, while the squippies just fill in the remaining gaps. Note that it does not use any tetrahedra (nor triangular prisms or any other "usual" small stuff).

Incidence matrix according to Dynkin symbol

FxFox5xFoFx xofox5oxofx&#zx   → all existing heights = 0

o....5o.... o....5o....     | 50  *  *  *   * |  1  2   2  1  0  0  0   0   0  0  0  0  0 |  2  1  2  1  1   2  0  0  0  0  0  0  0  0  0  0  0  0 |  1 1  1  2 0  0  0 0 0
.o...5.o... .o...5.o...     |  * 50  *  *   * |  0  0   2  0  1  2  1   0   0  0  0  0  0 |  0  0  1  2  0   2  2  1  1  0  0  0  0  0  0  0  0  0 |  1 0  2  1 1  0  0 0 0
..o..5..o.. ..o..5..o..     |  *  * 25  *   * |  0  0   0  2  0  0  0   4   0  0  0  0  0 |  0  0  0  0  1   4  0  0  0  2  2  0  0  0  0  0  0  0 |  0 0  2  2 0  1  0 0 0
...o.5...o. ...o.5...o.     |  *  *  * 25   * |  0  0   0  0  0  0  2   0   4  0  0  0  0 |  0  0  0  0  0   4  0  0  1  0  0  2  2  0  0  0  0  0 |  0 0  2  2 0  0  1 0 0
....o5....o ....o5....o     |  *  *  *  * 100 |  0  0   0  0  0  0  0   1   1  1  1  1  1 |  0  0  0  0  0   1  0  0  0  1  1  1  1  1  1  1  1  1 |  0 0  1  1 0  1  1 1 1
----------------------------+-----------------+-------------------------------------------+--------------------------------------------------------+-----------------------
..... x.... ..... .....     |  2  0  0  0   0 | 25  *   *  *  *  *  *   *   *  *  *  *  * |  2  0  0  0  1   0  0  0  0  0  0  0  0  0  0  0  0  0 |  0 1  0  2 0  0  0 0 0
..... ..... x.... .....     |  2  0  0  0   0 |  * 50   *  *  *  *  *   *   *  *  *  *  * |  1  1  1  0  0   0  0  0  0  0  0  0  0  0  0  0  0  0 |  1 1  0  1 0  0  0 0 0
oo...5oo... oo...5oo...&#x  |  1  1  0  0   0 |  *  * 100  *  *  *  *   *   *  *  *  *  * |  0  0  1  1  0   1  0  0  0  0  0  0  0  0  0  0  0  0 |  1 0  1  1 0  0  0 0 0
o.o..5o.o.. o.o..5o.o..&#x  |  1  0  1  0   0 |  *  *   * 50  *  *  *   *   *  *  *  *  * |  0  0  0  0  1   2  0  0  0  0  0  0  0  0  0  0  0  0 |  0 0  1  2 0  0  0 0 0
.x... ..... ..... .....     |  0  2  0  0   0 |  *  *   *  * 25  *  *   *   *  *  *  *  * |  0  0  0  0  0   0  2  0  1  0  0  0  0  0  0  0  0  0 |  0 0  2  0 1  0  0 0 0
..... ..... ..... .x...     |  0  2  0  0   0 |  *  *   *  *  * 50  *   *   *  *  *  *  * |  0  0  0  1  0   0  1  1  0  0  0  0  0  0  0  0  0  0 |  1 0  1  0 1  0  0 0 0
.o.o.5.o.o. .o.o.5.o.o.&#x  |  0  1  0  1   0 |  *  *   *  *  *  * 50   *   *  *  *  *  * |  0  0  0  0  0   2  0  0  1  0  0  0  0  0  0  0  0  0 |  0 0  2  1 0  0  0 0 0
..o.o5..o.o ..o.o5..o.o&#x  |  0  0  1  0   1 |  *  *   *  *  *  *  * 100   *  *  *  *  * |  0  0  0  0  0   1  0  0  0  1  1  0  0  0  0  0  0  0 |  0 0  1  1 0  1  0 0 0
...oo5...oo ...oo5...oo&#x  |  0  0  0  1   1 |  *  *   *  *  *  *  *   * 100  *  *  *  * |  0  0  0  0  0   1  0  0  0  0  0  1  1  0  0  0  0  0 |  0 0  1  1 0  0  1 0 0
....x ..... ..... .....     |  0  0  0  0   2 |  *  *   *  *  *  *  *   *   * 50  *  *  * |  0  0  0  0  0   0  0  0  0  0  0  1  0  1  1  1  0  0 |  0 0  1  0 0  0  1 1 1
..... ....x ..... .....     |  0  0  0  0   2 |  *  *   *  *  *  *  *   *   *  * 50  *  * |  0  0  0  0  0   0  0  0  0  1  0  0  0  1  0  0  1  1 |  0 0  0  1 0  1  0 1 1
..... ..... ....x .....     |  0  0  0  0   2 |  *  *   *  *  *  *  *   *   *  *  * 50  * |  0  0  0  0  0   0  0  0  0  0  0  0  1  0  1  0  1  0 |  0 0  0  1 0  0  1 1 0
..... ..... ..... ....x     |  0  0  0  0   2 |  *  *   *  *  *  *  *   *   *  *  *  * 50 |  0  0  0  0  0   0  0  0  0  0  1  0  0  0  0  1  0  1 |  0 0  1  0 0  1  0 0 1
----------------------------+-----------------+-------------------------------------------+--------------------------------------------------------+-----------------------
..... x.... x.... .....     |  4  0  0  0   0 |  2  2   0  0  0  0  0   0   0  0  0  0  0 | 25  *  *  *  *   *  *  *  *  *  *  *  *  *  *  *  *  * |  0 1  0  1 0  0  0 0 0
..... ..... x....5o....     |  5  0  0  0   0 |  0  5   0  0  0  0  0   0   0  0  0  0  0 |  * 10  *  *  *   *  *  *  *  *  *  *  *  *  *  *  *  * |  1 1  0  0 0  0  0 0 0
..... ..... xo... .....&#x  |  2  1  0  0   0 |  0  1   2  0  0  0  0   0   0  0  0  0  0 |  *  * 50  *  *   *  *  *  *  *  *  *  *  *  *  *  *  * |  1 0  0  1 0  0  0 0 0
..... ..... ..... ox...&#x  |  1  2  0  0   0 |  0  0   2  0  0  1  0   0   0  0  0  0  0 |  *  *  * 50  *   *  *  *  *  *  *  *  *  *  *  *  *  * |  1 0  1  0 0  0  0 0 0
..... x.o.. ..... .....&#x  |  2  0  1  0   0 |  1  0   0  2  0  0  0   0   0  0  0  0  0 |  *  *  *  * 25   *  *  *  *  *  *  *  *  *  *  *  *  * |  0 0  0  2 0  0  0 0 0
ooooo5ooooo ooooo5ooooo&#xr |  1  1  1  1   1 |  0  0   1  1  0  0  1   1   1  0  0  0  0 |  *  *  *  *  * 100  *  *  *  *  *  *  *  *  *  *  *  * |  0 0  1  1 0  0  0 0 0
.x... ..... ..... .x...     |  0  4  0  0   0 |  0  0   0  0  2  2  0   0   0  0  0  0  0 |  *  *  *  *  *   * 25  *  *  *  *  *  *  *  *  *  *  * |  0 0  1  0 1  0  0 0 0
..... ..... .o...5.x...     |  0  5  0  0   0 |  0  0   0  0  0  5  0   0   0  0  0  0  0 |  *  *  *  *  *   *  * 10  *  *  *  *  *  *  *  *  *  * |  1 0  0  0 1  0  0 0 0
.x.o. ..... ..... .....&#x  |  0  2  0  1   0 |  0  0   0  0  1  0  2   0   0  0  0  0  0 |  *  *  *  *  *   *  *  * 25  *  *  *  *  *  *  *  *  * |  0 0  2  0 0  0  0 0 0
..... ..o.x ..... .....&#x  |  0  0  1  0   2 |  0  0   0  0  0  0  0   2   0  0  1  0  0 |  *  *  *  *  *   *  *  *  * 50  *  *  *  *  *  *  *  * |  0 0  0  1 0  1  0 0 0
..... ..... ..... ..o.x&#x  |  0  0  1  0   2 |  0  0   0  0  0  0  0   2   0  0  0  0  1 |  *  *  *  *  *   *  *  *  *  * 50  *  *  *  *  *  *  * |  0 0  1  0 0  1  0 0 0
...ox ..... ..... .....&#x  |  0  0  0  1   2 |  0  0   0  0  0  0  0   0   2  1  0  0  0 |  *  *  *  *  *   *  *  *  *  *  * 50  *  *  *  *  *  * |  0 0  1  0 0  0  1 0 0
..... ..... ...ox .....&#x  |  0  0  0  1   2 |  0  0   0  0  0  0  0   0   2  0  0  1  0 |  *  *  *  *  *   *  *  *  *  *  *  * 50  *  *  *  *  * |  0 0  0  1 0  0  1 0 0
....x5....x ..... .....     |  0  0  0  0  10 |  0  0   0  0  0  0  0   0   0  5  5  0  0 |  *  *  *  *  *   *  *  *  *  *  *  *  * 10  *  *  *  * |  0 0  0  0 0  0  0 1 1
....x ..... ....x .....     |  0  0  0  0   4 |  0  0   0  0  0  0  0   0   0  2  0  2  0 |  *  *  *  *  *   *  *  *  *  *  *  *  *  * 25  *  *  * |  0 0  0  0 0  0  1 1 0
....x ..... ..... ....x     |  0  0  0  0   4 |  0  0   0  0  0  0  0   0   0  2  0  0  2 |  *  *  *  *  *   *  *  *  *  *  *  *  *  *  * 25  *  * |  0 0  1  0 0  0  0 0 1
..... ....x ....x .....     |  0  0  0  0   4 |  0  0   0  0  0  0  0   0   0  0  2  2  0 |  *  *  *  *  *   *  *  *  *  *  *  *  *  *  *  * 25  * |  0 0  0  1 0  0  0 1 0
..... ....x ..... ....x     |  0  0  0  0   4 |  0  0   0  0  0  0  0   0   0  0  2  0  2 |  *  *  *  *  *   *  *  *  *  *  *  *  *  *  *  *  * 25 |  0 0  0  0 0  1  0 0 1
----------------------------+-----------------+-------------------------------------------+--------------------------------------------------------+-----------------------
..... ..... xo...5ox...&#x  ♦  5  5  0  0   0 |  0  5  10  0  0  5  0   0   0  0  0  0  0 |  0  1  5  5  0   0  0  1  0  0  0  0  0  0  0  0  0  0 | 10 *  *  * *  *  * * *
..... x.... x....5o....     ♦ 10  0  0  0   0 |  5 10   0  0  0  0  0   0   0  0  0  0  0 |  5  2  0  0  0   0  0  0  0  0  0  0  0  0  0  0  0  0 |  * 5  *  * *  *  * * *
FxFox ..... ..... oxofx&#xt ♦  2  4  2  2   4 |  0  0   4  2  2  2  4   4   4  2  0  0  2 |  0  0  0  2  0   4  1  0  2  0  2  2  0  0  0  1  0  0 |  * * 25  * *  *  * * * (tower: 21435)
..... xFoFx xofox .....&#xt ♦  4  2  2  2   4 |  2  2   4  4  0  0  2   4   4  0  2  2  0 |  1  0  2  0  2   4  0  0  0  2  0  0  2  0  0  0  1  0 |  * *  * 25 *  *  * * * (tower: 12345)
.x... ..... .o...5.x...     ♦  0 10  0  0   0 |  0  0   0  0  5 10  0   0   0  0  0  0  0 |  0  0  0  0  0   0  5  2  0  0  0  0  0  0  0  0  0  0 |  * *  *  * 5  *  * * *
..... ..o.x ..... ..o.x&#x  ♦  0  0  1  0   4 |  0  0   0  0  0  0  0   4   0  0  2  0  2 |  0  0  0  0  0   0  0  0  0  2  2  0  0  0  0  0  0  1 |  * *  *  * * 25  * * *
...ox ..... ...ox .....&#x  ♦  0  0  0  1   4 |  0  0   0  0  0  0  0   0   4  2  0  2  0 |  0  0  0  0  0   0  0  0  0  0  0  2  2  0  1  0  0  0 |  * *  *  * *  * 25 * *
....x5....x ....x .....     ♦  0  0  0  0  20 |  0  0   0  0  0  0  0   0   0 10 10 10  0 |  0  0  0  0  0   0  0  0  0  0  0  0  0  2  5  0  5  0 |  * *  *  * *  *  * 5 *
....x5....x ..... ....x     ♦  0  0  0  0  20 |  0  0   0  0  0  0  0   0   0 10 10  0 10 |  0  0  0  0  0   0  0  0  0  0  0  0  0  2  0  5  0  5 |  * *  *  * *  *  * * 5

or
o....5o.... o....5o....     & | 100  *   * |  1   2   2   1   0   0   0 |  2  1   3  1   2   0   0  0  0  0 |  1  1  3  0  0
..o..5..o.. ..o..5..o..     & |   * 50   * |  0   0   0   2   4   0   0 |  0  0   0  1   4   2   2  0  0  0 |  0  0  4  1  0
....o5....o ....o5....o       |   *  * 100 |  0   0   0   0   2   2   2 |  0  0   0  0   1   2   2  1  2  2 |  0  0  2  2  2
------------------------------+------------+----------------------------+-----------------------------------+---------------
..... x.... ..... .....     & |   2  0   0 | 50   *   *   *   *   *   * |  2  0   0  1   0   0   0  0  0  0 |  0  1  2  0  0
..... ..... x.... .....     & |   2  0   0 |  * 100   *   *   *   *   * |  1  1   1  0   0   0   0  0  0  0 |  1  1  1  0  0
oo...5oo... oo...5oo...&#x    |   2  0   0 |  *   * 100   *   *   *   * |  0  0   2  0   1   0   0  0  0  0 |  1  0  2  0  0
o.o..5o.o.. o.o..5o.o..&#x  & |   1  1   0 |  *   *   * 100   *   *   * |  0  0   0  1   2   0   0  0  0  0 |  0  0  3  0  0
..o.o5..o.o ..o.o5..o.o&#x  & |   0  1   1 |  *   *   *   * 200   *   * |  0  0   0  0   1   1   1  0  0  0 |  0  0  2  1  0
....x ..... ..... .....     & |   0  0   2 |  *   *   *   *   * 100   * |  0  0   0  0   0   1   0  1  1  1 |  0  0  1  1  2
..... ..... ....x .....     & |   0  0   2 |  *   *   *   *   *   * 100 |  0  0   0  0   0   0   1  0  1  1 |  0  0  1  1  1
------------------------------+------------+----------------------------+-----------------------------------+---------------
..... x.... x.... .....     & |   4  0   0 |  2   2   0   0   0   0   0 | 50  *   *  *   *   *   *  *  *  * |  0  1  1  0  0
..... ..... x....5o....     & |   5  0   0 |  0   5   0   0   0   0   0 |  * 20   *  *   *   *   *  *  *  * |  1  1  0  0  0
..... ..... xo... .....&#x  & |   3  0   0 |  0   1   2   0   0   0   0 |  *  * 100  *   *   *   *  *  *  * |  1  0  1  0  0
..... x.o.. ..... .....&#x  & |   2  1   0 |  1   0   0   2   0   0   0 |  *  *   * 50   *   *   *  *  *  * |  0  0  2  0  0
ooooo5ooooo ooooo5ooooo&#xr   |   2  2   1 |  0   0   1   2   2   0   0 |  *  *   *  * 100   *   *  *  *  * |  0  0  2  0  0
..... ..o.x ..... .....&#x  & |   0  1   2 |  0   0   0   0   2   1   0 |  *  *   *  *   * 100   *  *  *  * |  0  0  1  1  0
..... ..... ..... ..o.x&#x  & |   0  1   2 |  0   0   0   0   2   0   1 |  *  *   *  *   *   * 100  *  *  * |  0  0  1  1  0
....x5....x ..... .....       |   0  0  10 |  0   0   0   0   0  10   0 |  *  *   *  *   *   *   * 10  *  * |  0  0  0  0  2
....x ..... ....x .....     & |   0  0   4 |  0   0   0   0   0   2   2 |  *  *   *  *   *   *   *  * 50  * |  0  0  0  1  1
....x ..... ..... ....x     & |   0  0   4 |  0   0   0   0   0   2   2 |  *  *   *  *   *   *   *  *  * 50 |  0  0  1  0  1
------------------------------+------------+----------------------------+-----------------------------------+---------------
..... ..... xo...5ox...&#x    ♦  10  0   0 |  0  10  10   0   0   0   0 |  0  2  10  0   0   0   0  0  0  0 | 10  *  *  *  *
..... x.... x....5o....     & ♦  10  0   0 |  5  10   0   0   0   0   0 |  5  2   0  0   0   0   0  0  0  0 |  * 10  *  *  *
FxFox ..... ..... oxofx&#xt & ♦   6  4   4 |  2   2   4   6   8   2   2 |  1  0   2  2   4   2   2  0  0  1 |  *  * 50  *  * (tower: 21435)
..... ..o.x ..... ..o.x&#x  & ♦   0  1   4 |  0   0   0   0   4   2   2 |  0  0   0  0   0   2   2  0  1  0 |  *  *  * 50  *
....x5....x ....x .....     & ♦   0  0  20 |  0   0   0   0   0  20  10 |  0  0   0  0   0   0   0  2  5  5 |  *  *  *  * 10