x3o   o3f f3o   o3x
                   
                   
   f3x       x3f   
                   
     o3F   F3o     
                   
                   
        x3x        

This multi-wedge clearly can be augmented by a ikepy to become the thawro-wedged 1/6-luna of ex. – However both, this polychoron and its augmentation, are not convex. This is because of the offending f-edges of the part of the lace city, which is Fo3oF&#f, which would show up in their hulls. (Just confer, how these f-edges could be recovered by incident pentagons within thawrorh, which then becomes convex again, for instance.)

Incidence matrix

6 *  * * * | 2  2  2  0  0 0 0 0  0  0  0 0  0 0 0 | 1 2 2  2  2 1 0 0  0  0  0 0 0  0  0  0  0 0 0  0 | 2 2  2 0 0  0 0 0  0 0  at x3x (of l.c.) ones
* 6  * * * | 0  2  0  2  2 1 0 0  0  0  0 0  0 0 0 | 0 1 0  2  2 0 1 1  2  2  2 0 0  0  0  0  0 0 0  0 | 1 1  2 1 1  2 0 0  0 0  at o3F (of l.c.) ones
* * 12 * * | 0  0  1  1  1 0 1 1  1  1  1 0  0 0 0 | 0 0 1  1  1 1 1 1  2  1  1 1 1  2  1  1  1 0 0  0 | 1 1  2 1 1  2 1 1  1 0  at f3x (of l.c.) ones
* *  * 6 * | 0  0  0  0  0 0 0 0  2  0  0 2  2 1 0 | 0 0 0  0  0 0 2 0  0  0  0 1 0  0  2  2  0 1 2  2 | 1 0  0 2 0  0 1 0  2 1  at x3o (of l.c.) ones
* *  * * 6 | 0  0  0  0  0 1 0 0  0  2  2 0  2 1 2 | 0 0 0  0  0 0 0 0  0  2  2 0 1  2  2  2  4 0 1  4 | 0 0  0 1 1  2 1 2  4 1  at o3f (of l.c.) ones
-----------+---------------------------------------+---------------------------------------------------+-----------------------
2 0  0 0 0 | 6  *  *  *  * * * *  *  *  * *  * * * | 1 1 1  0  0 0 0 0  0  0  0 0 0  0  0  0  0 0 0  0 | 2 1  0 0 0  0 0 0  0 0
1 1  0 0 0 | * 12  *  *  * * * *  *  *  * *  * * * | 0 1 0  1  1 0 0 0  0  0  0 0 0  0  0  0  0 0 0  0 | 1 1  1 0 0  0 0 0  0 0
1 0  1 0 0 | *  * 12  *  * * * *  *  *  * *  * * * | 0 0 1  1  1 1 0 0  0  0  0 0 0  0  0  0  0 0 0  0 | 1 1  2 0 0  0 0 0  0 0
0 1  1 0 0 | *  *  * 12  * * * *  *  *  * *  * * * | 0 0 0  1  0 0 1 0  1  1  0 0 0  0  0  0  0 0 0  0 | 1 0  1 1 0  1 0 0  0 0  of each thawro
0 1  1 0 0 | *  *  *  * 12 * * *  *  *  * *  * * * | 0 0 0  0  1 0 0 1  1  0  1 0 0  0  0  0  0 0 0  0 | 0 1  1 0 1  1 0 0  0 0  between thawroes
0 1  0 0 1 | *  *  *  *  * 6 * *  *  *  * *  * * * | 0 0 0  0  0 0 0 0  0  2  2 0 0  0  0  0  0 0 0  0 | 0 0  0 1 1  2 0 0  0 0  near
0 0  2 0 0 | *  *  *  *  * * 6 *  *  *  * *  * * * | 0 0 1  0  0 0 0 1  0  0  0 1 1  0  0  0  0 0 0  0 | 1 1  0 0 1  0 1 0  0 0  of each thawro
0 0  2 0 0 | *  *  *  *  * * * 6  *  *  * *  * * * | 0 0 0  0  0 1 0 0  2  0  0 0 0  2  0  0  0 0 0  0 | 0 0  2 0 0  2 0 1  0 0  between thawroes
0 0  1 1 0 | *  *  *  *  * * * * 12  *  * *  * * * | 0 0 0  0  0 0 1 0  0  0  0 1 0  0  1  1  0 0 0  0 | 1 0  0 1 0  0 1 0  1 0
0 0  1 0 1 | *  *  *  *  * * * *  * 12  * *  * * * | 0 0 0  0  0 0 0 0  0  1  0 0 0  1  1  0  1 0 0  0 | 0 0  0 1 0  1 0 1  1 0  peppy lacings (near)
0 0  1 0 1 | *  *  *  *  * * * *  *  * 12 *  * * * | 0 0 0  0  0 0 0 0  0  0  1 0 1  1  0  1  1 0 0  0 | 0 0  0 0 1  1 1 1  1 0  far
0 0  0 2 0 | *  *  *  *  * * * *  *  *  * 6  * * * | 0 0 0  0  0 0 1 0  0  0  0 0 0  0  0  0  0 1 1  0 | 1 0  0 1 0  0 0 0  0 1
0 0  0 1 1 | *  *  *  *  * * * *  *  *  * * 12 * * | 0 0 0  0  0 0 0 0  0  0  0 0 0  0  1  0  0 0 1  1 | 0 0  0 1 0  0 0 0  1 1  near
0 0  0 1 1 | *  *  *  *  * * * *  *  *  * *  * 6 * | 0 0 0  0  0 0 0 0  0  0  0 0 0  0  0  2  0 0 0  2 | 0 0  0 0 0  0 1 0  2 1  far
0 0  0 0 2 | *  *  *  *  * * * *  *  *  * *  * * 6 | 0 0 0  0  0 0 0 0  0  0  0 0 0  0  0  0  2 0 0  2 | 0 0  0 0 0  0 0 1  2 1
-----------+---------------------------------------+---------------------------------------------------+-----------------------
6 0  0 0 0 | 6  0  0  0  0 0 0 0  0  0  0 0  0 0 0 | 1 * *  *  * * * *  *  *  * * *  *  *  *  * * *  * | 2 0  0 0 0  0 0 0  0 0
2 1  0 0 0 | 1  2  0  0  0 0 0 0  0  0  0 0  0 0 0 | * 6 *  *  * * * *  *  *  * * *  *  *  *  * * *  * | 1 1  0 0 0  0 0 0  0 0
2 0  2 0 0 | 1  0  2  0  0 0 1 0  0  0  0 0  0 0 0 | * * 6  *  * * * *  *  *  * * *  *  *  *  * * *  * | 1 1  0 0 0  0 0 0  0 0
1 1  1 0 0 | 0  1  1  1  0 0 0 0  0  0  0 0  0 0 0 | * * * 12  * * * *  *  *  * * *  *  *  *  * * *  * | 1 0  1 0 0  0 0 0  0 0  of each thawro
1 1  1 0 0 | 0  1  1  0  1 0 0 0  0  0  0 0  0 0 0 | * * *  * 12 * * *  *  *  * * *  *  *  *  * * *  * | 0 1  1 0 0  0 0 0  0 0  between thawroes
1 0  2 0 0 | 0  0  2  0  0 0 0 1  0  0  0 0  0 0 0 | * * *  *  * 6 * *  *  *  * * *  *  *  *  * * *  * | 0 0  2 0 0  0 0 0  0 0
0 1  2 2 0 | 0  0  0  2  0 0 0 0  2  0  0 1  0 0 0 | * * *  *  * * 6 *  *  *  * * *  *  *  *  * * *  * | 1 0  0 1 0  0 0 0  0 0
0 1  2 0 0 | 0  0  0  0  2 0 1 0  0  0  0 0  0 0 0 | * * *  *  * * * 6  *  *  * * *  *  *  *  * * *  * | 0 1  0 0 1  0 0 0  0 0  at squippy (para)
0 1  2 0 0 | 0  0  0  1  1 0 0 1  0  0  0 0  0 0 0 | * * *  *  * * * * 12  *  * * *  *  *  *  * * *  * | 0 0  1 0 0  1 0 0  0 0
0 1  1 0 1 | 0  0  0  1  0 1 0 0  0  1  0 0  0 0 0 | * * *  *  * * * *  * 12  * * *  *  *  *  * * *  * | 0 0  0 1 0  1 0 0  0 0
0 1  1 0 1 | 0  0  0  0  1 1 0 0  0  0  1 0  0 0 0 | * * *  *  * * * *  *  * 12 * *  *  *  *  * * *  * | 0 0  0 0 1  1 0 0  0 0
0 0  2 1 0 | 0  0  0  0  0 0 1 0  2  0  0 0  0 0 0 | * * *  *  * * * *  *  *  * 6 *  *  *  *  * * *  * | 1 0  0 0 0  0 1 0  0 0
0 0  2 0 1 | 0  0  0  0  0 0 1 0  0  0  2 0  0 0 0 | * * *  *  * * * *  *  *  * * 6  *  *  *  * * *  * | 0 0  0 0 1  0 1 0  0 0  connecting one thawro to far f3o vertex
0 0  2 0 1 | 0  0  0  0  0 0 0 1  0  1  1 0  0 0 0 | * * *  *  * * * *  *  *  * * * 12  *  *  * * *  * | 0 0  0 0 0  1 0 1  0 0  between thawroes
0 0  1 1 1 | 0  0  0  0  0 0 0 0  1  1  0 0  1 0 0 | * * *  *  * * * *  *  *  * * *  * 12  *  * * *  * | 0 0  0 1 0  0 0 0  1 0  with near o3f vertex
0 0  1 1 1 | 0  0  0  0  0 0 0 0  1  0  1 0  0 1 0 | * * *  *  * * * *  *  *  * * *  *  * 12  * * *  * | 0 0  0 0 0  0 1 0  1 0  with far f3o vertex
0 0  1 0 2 | 0  0  0  0  0 0 0 0  0  1  1 0  0 0 1 | * * *  *  * * * *  *  *  * * *  *  *  * 12 * *  * | 0 0  0 0 0  0 0 1  1 0
0 0  0 3 0 | 0  0  0  0  0 0 0 0  0  0  0 3  0 0 0 | * * *  *  * * * *  *  *  * * *  *  *  *  * 2 *  * | 1 0  0 0 0  0 0 0  0 1
0 0  0 2 1 | 0  0  0  0  0 0 0 0  0  0  0 1  2 0 0 | * * *  *  * * * *  *  *  * * *  *  *  *  * * 6  * | 0 0  0 1 0  0 0 0  0 1
0 0  0 1 2 | 0  0  0  0  0 0 0 0  0  0  0 0  1 1 1 | * * *  *  * * * *  *  *  * * *  *  *  *  * * * 12 | 0 0  0 0 0  0 0 0  1 1
-----------+---------------------------------------+---------------------------------------------------+-----------------------
6 3  6 3 0 | 6  6  6  6  0 0 3 0  6  0  0 3  0 0 0 | 1 3 3  6  0 0 3 0  0  0  0 3 0  0  0  0  0 1 0  0 | 2 *  * * *  * * *  * *  thawro
2 1  2 0 0 | 1  2  2  0  2 0 1 0  0  0  0 0  0 0 0 | 0 1 1  0  2 0 0 1  0  0  0 0 0  0  0  0  0 0 0  0 | * 6  * * *  * * *  * *  squippy
1 1  2 0 0 | 0  1  2  1  1 0 0 1  0  0  0 0  0 0 0 | 0 0 0  1  1 1 0 0  1  0  0 0 0  0  0  0  0 0 0  0 | * * 12 * *  * * *  * *  tet
0 1  2 2 1 | 0  0  0  2  0 1 0 0  2  2  0 1  2 0 0 | 0 0 0  0  0 0 1 0  0  2  0 0 0  0  2  0  0 0 1  0 | * *  * 6 *  * * *  * *  peppy
0 1  2 0 1 | 0  0  0  0  2 1 1 0  0  0  2 0  0 0 0 | 0 0 0  0  0 0 0 1  0  0  2 0 1  0  0  0  0 0 0  0 | * *  * * 6  * * *  * *  tet  at thawro's horizontal {4}-edge
0 1  2 0 1 | 0  0  0  1  1 1 0 1  0  1  1 0  0 0 0 | 0 0 0  0  0 0 0 0  1  1  1 0 0  1  0  0  0 0 0  0 | * *  * * * 12 * *  * *  tet  with inter-thawro's upper {4}-vertex edge
0 0  2 1 1 | 0  0  0  0  0 0 1 0  2  0  2 0  0 1 0 | 0 0 0  0  0 0 0 0  0  0  0 1 1  0  0  2  0 0 0  0 | * *  * * *  * 6 *  * *  tet  with far f3o vertex
0 0  2 0 2 | 0  0  0  0  0 0 0 1  0  2  2 0  0 0 1 | 0 0 0  0  0 0 0 0  0  0  0 0 0  2  0  0  2 0 0  0 | * *  * * *  * * 6  * *  tet  between thawroes
0 0  1 1 2 | 0  0  0  0  0 0 0 0  1  1  1 0  1 1 1 | 0 0 0  0  0 0 0 0  0  0  0 0 0  0  1  1  1 0 0  1 | * *  * * *  * * * 12 *  tet
0 0  0 6 6 | 0  0  0  0  0 0 0 0  0  0  0 6 12 6 6 | 0 0 0  0  0 0 0 0  0  0  0 0 0  0  0  0  0 2 6 12 | * *  * * *  * * *  * 1  ike