By the very definition of Waterman polytopes, not necessarily all vertices are on the same sphere. However in here both the 16 d-tes vertices and the 192 (q,q,x)-prit vertices belong to the same radius.

Incidence matrix according to Dynkin symbol

oq3oq3oo4dx&#zh   → height = 0
                    where d = 3 (pseudo)
(tegum sum of d-tes and (q,q,x)-prit)

o.3o.3o.4o.     | 16   * ♦  12  0   0   0 |  6  12  0  0  0  0 |  4  4  0 0
.o3.o3.o4.o     |  * 192 |   1  1   2   2 |  1   2  2  1  2  1 |  2  1  1 1
----------------+--------+----------------+--------------------+-----------
oo3oo3oo4oo&#h  |  1   1 | 192  *   *   * |  1   2  0  0  0  0 |  2  1  0 0
.q .. .. ..     |  0   2 |   * 96   *   * |  1   0  2  0  0  0 |  2  0  1 0
.. .q .. ..     |  0   2 |   *  * 192   * |  0   1  0  1  1  0 |  1  1  0 1
.. .. .. .x     |  0   2 |   *  *   * 192 |  0   0  1  0  1  1 |  1  0  1 1
----------------+--------+----------------+--------------------+-----------
oq .. .. ..&#h  |  1   2 |   2  1   0   0 | 96   *  *  *  *  * |  2  0  0 0
.. oq .. ..&#h  |  1   2 |   2  0   1   0 |  * 192  *  *  *  * |  1  1  0 0
.q .. .. .x     |  0   4 |   0  2   0   2 |  *   * 96  *  *  * |  1  0  1 0
.. .q3.o ..     |  0   3 |   0  0   3   0 |  *   *  * 64  *  * |  0  1  0 1
.. .q .. .x     |  0   4 |   0  0   2   2 |  *   *  *  * 96  * |  1  0  0 1
.. .. .o4.x     |  0   4 |   0  0   0   4 |  *   *  *  *  * 48 |  0  0  1 1
----------------+--------+----------------+--------------------+-----------
oq3oq .. dx&#zh ♦  2  12 |  12  6   6   6 |  6   6  3  0  3  0 | 32  *  * *
.. oq3oo ..&#h  ♦  1   3 |   3  0   3   0 |  0   3  0  1  0  0 |  * 64  * *
.q .. .o4.x     ♦  0   8 |   0  4   0   8 |  0   0  4  0  0  2 |  *  * 24 *
.. .q3.o4.x     ♦  0  24 |   0  0  24  24 |  0   0  0  8 12  6 |  *  *  * 8