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

Incidence matrix according to Dynkin symbol

β5/2o5β

both( .   . . ) | 60 |  2  2  2 |  1  1  4
----------------+----+----------+---------
both( s   2 s ) |  2 | 60  *  * |  0  0  2
sefa( β5/2o . ) |  2 |  * 60  * |  1  0  1
sefa( .   o5β ) |  2 |  *  * 60 |  0  1  1
----------------+----+----------+---------
      β5/2o .   ♦  5 |  0  5  0 | 12  *  *
      .   o5β   ♦  5 |  0  0  5 |  * 12  *
sefa( β5/2o5β ) |  4 |  2  1  1 |  *  * 60

starting figure: x5/2o5x