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It should be noted here, that the squares of the components (raded-1-10-5-5) are situated fully within the mirror planes underneath the edges of doe. Accordingly this (centrally hollow) external blend from 12 such self-intersecting polyhedra can be constructed, blending out half of their squares.

Incidence matrix according to Dynkin symbol

reduced( Ffoo5xxux5/2ofxx&#zx , by 12{10/2} )   → heights = 0

o...5o...5/2o...      | 60   *  *  * |  2   2  0   0  0  0  0 |  1  2  1  0  [4,5,4,5/2]
.o..5.o..5/2.o..      |  * 120  *  * |  0   1  1   1  0  0  0 |  0  1  1  1  [4,5,6]
..o.5..o.5/2..o.      |  *   * 60  * |  0   0  0   2  1  1  0 |  0  0  2  2  [52,62]
...o5...o5/2...o      |  *   *  * 20 |  0   0  0   0  0  3  3 |  0  0  0  6  [66]/2
----------------------+--------------+------------------------+------------
.... x...   ....      |  2   0  0  0 | 60   *  *   *  *  *  * |  1  1  0  0
oo..5oo..5/2oo..&#x   |  1   1  0  0 |  * 120  *   *  *  *  * |  0  1  1  0
.... .x..   ....      |  0   2  0  0 |  *   * 60   *  *  *  * |  0  1  0  1
.oo.5.oo.5/2.oo.&#x   |  0   1  1  0 |  *   *  * 120  *  *  * |  0  0  1  1
.... ....   ..x.      |  0   0  2  0 |  *   *  *   * 30  *  * |  0  0  2  0
..oo5..oo5/2..oo&#x   |  0   0  1  1 |  *   *  *   *  * 60  * |  0  0  0  2
.... ...x   ....    & |  0   0  0  2 |  *   *  *   *  *  * 30 |  0  0  0  2
----------------------+--------------+------------------------+------------
.... x...5/2o...      |  5   0  0  0 |  5   0  0   0  0  0  0 | 12  *  *  *
.... xx..   ....&#x   |  2   2  0  0 |  1   2  1   0  0  0  0 |  * 60  *  *
.... ....   ofx.&#xt  |  1   2  2  0 |  0   2  0   2  1  0  0 |  *  * 60  *
.... .xux   ....&#xt  |  0   2  2  2 |  0   0  1   2  0  2  1 |  *  *  * 60