By the very definition of Waterman polytopes, not necessarily all vertices are on the same sphere. In here the 48 maximal ones (q-thex vertices) have a circumradius of sqrt(5) = 2.236068, while the other 16 vertices (u-tes ones) only are at an radius of 2.

Incidence matrix according to Dynkin symbol

((oq3oq3oo4uo))&#zh   → height = 0, where u = 2 (pseudo)
(tegum sum of q-thex and u-tes)

  o.3o.3o.4o.       | 16  * |  12  0  0 |  6  12  0 |  4  4 0  u-tes-vertices
  .o3.o3.o4.o       |  * 48 |   4  1  4 |  4   8  4 |  4  4 1  q-thex-vertices
--------------------+-------+-----------+-----------+--------
  oo3oo3oo4oo  &#h  |  1  1 | 192  *  * |  1   2  0 |  2  1 0  h
  .q .. .. ..       |  0  2 |   * 24  * |  4   0  0 |  4  0 0  q
  .. .q .. ..       |  0  2 |   *  * 96 |  0   2  2 |  1  2 1  q
--------------------+-------+-----------+-----------+--------
  oq .. .. ..  &#h  |  1  2 |   2  1  0 | 96   *  * |  2  0 0
  .. oq .. ..  &#h  |  1  2 |   2  0  1 |  * 192  * |  1  1 0
  .. .q3.o ..       |  0  3 |   0  0  3 |  *   * 64 |  0  1 1
--------------------+-------+-----------+-----------+--------
((oq3oq .. uo))&#zh |  2  6 |  12  3  3 |  6   6  0 | 32  * *  hexagonal bipyramid
  .. oq3oo ..  &#h  |  1  3 |   3  0  3 |  0   3  1 |  * 64 *  trigonal pyramid
  .. .q3.o4.o       |  0  6 |   0  0 12 |  0   0  8 |  *  * 8  q-oct