No uniform realisation is possible. Sets of 6 tetragons each build equatorial rings.

Incidence matrix according to Dynkin symbol

β5/2x5β

both( .   . . ) | 120 |  1  1  1  1 |  1  1  2
----------------+-----+-------------+---------
both( .   x . ) |   2 | 60  *  *  * |  1  1  0
both( s   2 s ) |   2 |  * 60  *  * |  0  0  2
sefa( β5/2x . ) |   2 |  *  * 60  * |  1  0  1
sefa( .   x5β ) |   2 |  *  *  * 60 |  0  1  1
----------------+-----+-------------+---------
      β5/2x .   ♦  10 |  5  0  5  0 | 12  *  *
      .   x5β   ♦  10 |  5  0  0  5 |  * 12  *
sefa( β5/2x5β ) |   4 |  0  2  1  1 |  *  * 60

starting figure: x5/2x5x