o3o        
                   
x3o   o3f f3o   o3x
                   
                   
   f3x       x3f   
                   
     o3F   F3o     
                   
                   
        x3x        

Sadly this luna would not be convex. This is because the part of the lace city, which is Fo3oF&#f, would feature its f-edges to the convex hull. – These f-edges are covered differently, by according incident pentagons each, both at the thawro-wedged 2/6-luna of ex and at the thawrorh, which thus become convex again.

For sure this polychoron could be diminished at the top vertex of the lace city, cutting off an ikepy. This then would result in the multi-wedge dim. thawro-wedged 1/6-luna of ex (which so wouldn't be convex for the same reason).

Incidence matrix

6 *  * * * * | 2  2  2  0  0 0 0 0  0  0  0 0  0 0 0 0 0 | 1 2 2  2  2 1 0 0  0  0  0 0 0  0  0  0  0 0 0 0  0  0 0 0 | 2 2  2 0 0  0 0 0  0 0 0  0  at x3x (of l.c.) ones
* 6  * * * * | 0  2  0  2  2 1 0 0  0  0  0 0  0 0 0 0 0 | 0 1 0  2  2 0 1 1  2  2  2 0 0  0  0  0  0 0 0 0  0  0 0 0 | 1 1  2 1 1  2 0 0  0 0 0  0  at o3F (of l.c.) ones
* * 12 * * * | 0  0  1  1  1 0 1 1  1  1  1 0  0 0 0 0 0 | 0 0 1  1  1 1 1 1  2  1  1 1 1  2  1  1  1 0 0 0  0  0 0 0 | 1 1  2 1 1  2 1 1  1 0 0  0  at f3x (of l.c.) ones
* *  * 6 * * | 0  0  0  0  0 0 0 0  2  0  0 2  2 1 1 0 0 | 0 0 0  0  0 0 2 0  0  0  0 1 0  0  2  2  0 1 2 2  2  2 1 0 | 1 0  0 2 0  0 1 0  2 1 2  2  at x3o (of l.c.) ones
* *  * * 6 * | 0  0  0  0  0 1 0 0  0  2  2 0  2 1 0 2 1 | 0 0 0  0  0 0 0 0  0  2  2 0 1  2  2  2  4 0 1 0  4  2 1 2 | 0 0  0 1 1  2 1 2  4 0 1  4  at o3f (of l.c.) ones
* *  * * * 1 | 0  0  0  0  0 0 0 0  0  0  0 0  0 0 6 0 6 | 0 0 0  0  0 0 0 0  0  0  0 0 0  0  0  0  0 0 0 6  0 12 6 6 | 0 0  0 0 0  0 0 0  0 2 6 12  at o3o (of l.c.) one
-------------+-------------------------------------------+------------------------------------------------------------+----------------------------
2 0  0 0 0 0 | 6  *  *  *  * * * *  *  *  * *  * * * * * | 1 1 1  0  0 0 0 0  0  0  0 0 0  0  0  0  0 0 0 0  0  0 0 0 | 2 1  0 0 0  0 0 0  0 0 0  0
1 1  0 0 0 0 | * 12  *  *  * * * *  *  *  * *  * * * * * | 0 1 0  1  1 0 0 0  0  0  0 0 0  0  0  0  0 0 0 0  0  0 0 0 | 1 1  1 0 0  0 0 0  0 0 0  0
1 0  1 0 0 0 | *  * 12  *  * * * *  *  *  * *  * * * * * | 0 0 1  1  1 1 0 0  0  0  0 0 0  0  0  0  0 0 0 0  0  0 0 0 | 1 1  2 0 0  0 0 0  0 0 0  0
0 1  1 0 0 0 | *  *  * 12  * * * *  *  *  * *  * * * * * | 0 0 0  1  0 0 1 0  1  1  0 0 0  0  0  0  0 0 0 0  0  0 0 0 | 1 0  1 1 0  1 0 0  0 0 0  0  of each thawro
0 1  1 0 0 0 | *  *  *  * 12 * * *  *  *  * *  * * * * * | 0 0 0  0  1 0 0 1  1  0  1 0 0  0  0  0  0 0 0 0  0  0 0 0 | 0 1  1 0 1  1 0 0  0 0 0  0  between thawroes
0 1  0 0 1 0 | *  *  *  *  * 6 * *  *  *  * *  * * * * * | 0 0 0  0  0 0 0 0  0  2  2 0 0  0  0  0  0 0 0 0  0  0 0 0 | 0 0  0 1 1  2 0 0  0 0 0  0  near
0 0  2 0 0 0 | *  *  *  *  * * 6 *  *  *  * *  * * * * * | 0 0 1  0  0 0 0 1  0  0  0 1 1  0  0  0  0 0 0 0  0  0 0 0 | 1 1  0 0 1  0 1 0  0 0 0  0  of each thawro
0 0  2 0 0 0 | *  *  *  *  * * * 6  *  *  * *  * * * * * | 0 0 0  0  0 1 0 0  2  0  0 0 0  2  0  0  0 0 0 0  0  0 0 0 | 0 0  2 0 0  2 0 1  0 0 0  0  between thawroes
0 0  1 1 0 0 | *  *  *  *  * * * * 12  *  * *  * * * * * | 0 0 0  0  0 0 1 0  0  0  0 1 0  0  1  1  0 0 0 0  0  0 0 0 | 1 0  0 1 0  0 1 0  1 0 0  0
0 0  1 0 1 0 | *  *  *  *  * * * *  * 12  * *  * * * * * | 0 0 0  0  0 0 0 0  0  1  0 0 0  1  1  0  1 0 0 0  0  0 0 0 | 0 0  0 1 0  1 0 1  1 0 0  0  peppy lacings (near)
0 0  1 0 1 0 | *  *  *  *  * * * *  *  * 12 *  * * * * * | 0 0 0  0  0 0 0 0  0  0  1 0 1  1  0  1  1 0 0 0  0  0 0 0 | 0 0  0 0 1  1 1 1  1 0 0  0  far
0 0  0 2 0 0 | *  *  *  *  * * * *  *  *  * 6  * * * * * | 0 0 0  0  0 0 1 0  0  0  0 0 0  0  0  0  0 1 1 1  0  0 0 0 | 1 0  0 1 0  0 0 0  0 1 1  0
0 0  0 1 1 0 | *  *  *  *  * * * *  *  *  * * 12 * * * * | 0 0 0  0  0 0 0 0  0  0  0 0 0  0  1  0  0 0 1 0  1  1 0 0 | 0 0  0 1 0  0 0 0  1 0 1  1  near
0 0  0 1 1 0 | *  *  *  *  * * * *  *  *  * *  * 6 * * * | 0 0 0  0  0 0 0 0  0  0  0 0 0  0  0  2  0 0 0 0  2  0 1 0 | 0 0  0 0 0  0 1 0  2 0 0  2  far
0 0  0 1 0 1 | *  *  *  *  * * * *  *  *  * *  * * 6 * * | 0 0 0  0  0 0 0 0  0  0  0 0 0  0  0  0  0 0 0 2  0  2 1 0 | 0 0  0 0 0  0 0 0  0 1 2  2
0 0  0 0 2 0 | *  *  *  *  * * * *  *  *  * *  * * * 6 * | 0 0 0  0  0 0 0 0  0  0  0 0 0  0  0  0  2 0 0 0  2  0 0 1 | 0 0  0 0 0  0 0 1  2 0 0  2
0 0  0 0 1 1 | *  *  *  *  * * * *  *  *  * *  * * * * 6 | 0 0 0  0  0 0 0 0  0  0  0 0 0  0  0  0  0 0 0 0  0  2 1 2 | 0 0  0 0 0  0 0 0  0 0 1  4
-------------+-------------------------------------------+------------------------------------------------------------+----------------------------
6 0  0 0 0 0 | 6  0  0  0  0 0 0 0  0  0  0 0  0 0 0 0 0 | 1 * *  *  * * * *  *  *  * * *  *  *  *  * * * *  *  * * * | 2 0  0 0 0  0 0 0  0 0 0  0
2 1  0 0 0 0 | 1  2  0  0  0 0 0 0  0  0  0 0  0 0 0 0 0 | * 6 *  *  * * * *  *  *  * * *  *  *  *  * * * *  *  * * * | 1 1  0 0 0  0 0 0  0 0 0  0
2 0  2 0 0 0 | 1  0  2  0  0 0 1 0  0  0  0 0  0 0 0 0 0 | * * 6  *  * * * *  *  *  * * *  *  *  *  * * * *  *  * * * | 1 1  0 0 0  0 0 0  0 0 0  0
1 1  1 0 0 0 | 0  1  1  1  0 0 0 0  0  0  0 0  0 0 0 0 0 | * * * 12  * * * *  *  *  * * *  *  *  *  * * * *  *  * * * | 1 0  1 0 0  0 0 0  0 0 0  0  of each thawro
1 1  1 0 0 0 | 0  1  1  0  1 0 0 0  0  0  0 0  0 0 0 0 0 | * * *  * 12 * * *  *  *  * * *  *  *  *  * * * *  *  * * * | 0 1  1 0 0  0 0 0  0 0 0  0  between thawroes
1 0  2 0 0 0 | 0  0  2  0  0 0 0 1  0  0  0 0  0 0 0 0 0 | * * *  *  * 6 * *  *  *  * * *  *  *  *  * * * *  *  * * * | 0 0  2 0 0  0 0 0  0 0 0  0
0 1  2 2 0 0 | 0  0  0  2  0 0 0 0  2  0  0 1  0 0 0 0 0 | * * *  *  * * 6 *  *  *  * * *  *  *  *  * * * *  *  * * * | 1 0  0 1 0  0 0 0  0 0 0  0
0 1  2 0 0 0 | 0  0  0  0  2 0 1 0  0  0  0 0  0 0 0 0 0 | * * *  *  * * * 6  *  *  * * *  *  *  *  * * * *  *  * * * | 0 1  0 0 1  0 0 0  0 0 0  0  at squippy (para)
0 1  2 0 0 0 | 0  0  0  1  1 0 0 1  0  0  0 0  0 0 0 0 0 | * * *  *  * * * * 12  *  * * *  *  *  *  * * * *  *  * * * | 0 0  1 0 0  1 0 0  0 0 0  0
0 1  1 0 1 0 | 0  0  0  1  0 1 0 0  0  1  0 0  0 0 0 0 0 | * * *  *  * * * *  * 12  * * *  *  *  *  * * * *  *  * * * | 0 0  0 1 0  1 0 0  0 0 0  0
0 1  1 0 1 0 | 0  0  0  0  1 1 0 0  0  0  1 0  0 0 0 0 0 | * * *  *  * * * *  *  * 12 * *  *  *  *  * * * *  *  * * * | 0 0  0 0 1  1 0 0  0 0 0  0
0 0  2 1 0 0 | 0  0  0  0  0 0 1 0  2  0  0 0  0 0 0 0 0 | * * *  *  * * * *  *  *  * 6 *  *  *  *  * * * *  *  * * * | 1 0  0 0 0  0 1 0  0 0 0  0
0 0  2 0 1 0 | 0  0  0  0  0 0 1 0  0  0  2 0  0 0 0 0 0 | * * *  *  * * * *  *  *  * * 6  *  *  *  * * * *  *  * * * | 0 0  0 0 1  0 1 0  0 0 0  0  connecting one thawro to far f3o vertex
0 0  2 0 1 0 | 0  0  0  0  0 0 0 1  0  1  1 0  0 0 0 0 0 | * * *  *  * * * *  *  *  * * * 12  *  *  * * * *  *  * * * | 0 0  0 0 0  1 0 1  0 0 0  0  between thawroes
0 0  1 1 1 0 | 0  0  0  0  0 0 0 0  1  1  0 0  1 0 0 0 0 | * * *  *  * * * *  *  *  * * *  * 12  *  * * * *  *  * * * | 0 0  0 1 0  0 0 0  1 0 0  0  with near o3f vertex
0 0  1 1 1 0 | 0  0  0  0  0 0 0 0  1  0  1 0  0 1 0 0 0 | * * *  *  * * * *  *  *  * * *  *  * 12  * * * *  *  * * * | 0 0  0 0 0  0 1 0  1 0 0  0  with far f3o vertex
0 0  1 0 2 0 | 0  0  0  0  0 0 0 0  0  1  1 0  0 0 0 1 0 | * * *  *  * * * *  *  *  * * *  *  *  * 12 * * *  *  * * * | 0 0  0 0 0  0 0 1  1 0 0  0
0 0  0 3 0 0 | 0  0  0  0  0 0 0 0  0  0  0 3  0 0 0 0 0 | * * *  *  * * * *  *  *  * * *  *  *  *  * 2 * *  *  * * * | 1 0  0 0 0  0 0 0  0 1 0  0
0 0  0 2 1 0 | 0  0  0  0  0 0 0 0  0  0  0 1  2 0 0 0 0 | * * *  *  * * * *  *  *  * * *  *  *  *  * * 6 *  *  * * * | 0 0  0 1 0  0 0 0  0 0 1  0
0 0  0 2 0 1 | 0  0  0  0  0 0 0 0  0  0  0 1  0 0 2 0 0 | * * *  *  * * * *  *  *  * * *  *  *  *  * * * 6  *  * * * | 0 0  0 0 0  0 0 0  0 1 1  0
0 0  0 1 2 0 | 0  0  0  0  0 0 0 0  0  0  0 0  1 1 0 1 0 | * * *  *  * * * *  *  *  * * *  *  *  *  * * * * 12  * * * | 0 0  0 0 0  0 0 0  1 0 0  1
0 0  0 1 1 1 | 0  0  0  0  0 0 0 0  0  0  0 0  1 0 1 0 1 | * * *  *  * * * *  *  *  * * *  *  *  *  * * * *  * 12 * * | 0 0  0 0 0  0 0 0  0 0 1  1  near
0 0  0 1 1 1 | 0  0  0  0  0 0 0 0  0  0  0 0  0 1 1 0 1 | * * *  *  * * * *  *  *  * * *  *  *  *  * * * *  *  * 6 * | 0 0  0 0 0  0 0 0  0 0 0  2  far
0 0  0 0 2 1 | 0  0  0  0  0 0 0 0  0  0  0 0  0 0 0 1 2 | * * *  *  * * * *  *  *  * * *  *  *  *  * * * *  *  * * 6 | 0 0  0 0 0  0 0 0  0 0 0  2
-------------+-------------------------------------------+------------------------------------------------------------+----------------------------
6 3  6 3 0 0 | 6  6  6  6  0 0 3 0  6  0  0 3  0 0 0 0 0 | 1 3 3  6  0 0 3 0  0  0  0 3 0  0  0  0  0 1 0 0  0  0 0 0 | 2 *  * * *  * * *  * * *  *  thawro
2 1  2 0 0 0 | 1  2  2  0  2 0 1 0  0  0  0 0  0 0 0 0 0 | 0 1 1  0  2 0 0 1  0  0  0 0 0  0  0  0  0 0 0 0  0  0 0 0 | * 6  * * *  * * *  * * *  *  squippy
1 1  2 0 0 0 | 0  1  2  1  1 0 0 1  0  0  0 0  0 0 0 0 0 | 0 0 0  1  1 1 0 0  1  0  0 0 0  0  0  0  0 0 0 0  0  0 0 0 | * * 12 * *  * * *  * * *  *  tet
0 1  2 2 1 0 | 0  0  0  2  0 1 0 0  2  2  0 1  2 0 0 0 0 | 0 0 0  0  0 0 1 0  0  2  0 0 0  0  2  0  0 0 1 0  0  0 0 0 | * *  * 6 *  * * *  * * *  *  peppy
0 1  2 0 1 0 | 0  0  0  0  2 1 1 0  0  0  2 0  0 0 0 0 0 | 0 0 0  0  0 0 0 1  0  0  2 0 1  0  0  0  0 0 0 0  0  0 0 0 | * *  * * 6  * * *  * * *  *  tet  at thawro's horizontal {4}-edge
0 1  2 0 1 0 | 0  0  0  1  1 1 0 1  0  1  1 0  0 0 0 0 0 | 0 0 0  0  0 0 0 0  1  1  1 0 0  1  0  0  0 0 0 0  0  0 0 0 | * *  * * * 12 * *  * * *  *  tet  with inter-thawro's upper {4}-vertex edge
0 0  2 1 1 0 | 0  0  0  0  0 0 1 0  2  0  2 0  0 1 0 0 0 | 0 0 0  0  0 0 0 0  0  0  0 1 1  0  0  2  0 0 0 0  0  0 0 0 | * *  * * *  * 6 *  * * *  *  tet  with far f3o vertex
0 0  2 0 2 0 | 0  0  0  0  0 0 0 1  0  2  2 0  0 0 0 1 0 | 0 0 0  0  0 0 0 0  0  0  0 0 0  2  0  0  2 0 0 0  0  0 0 0 | * *  * * *  * * 6  * * *  *  tet  between thawroes
0 0  1 1 2 0 | 0  0  0  0  0 0 0 0  1  1  1 0  1 1 0 1 0 | 0 0 0  0  0 0 0 0  0  0  0 0 0  0  1  1  1 0 0 0  1  0 0 0 | * *  * * *  * * * 12 * *  *  tet
0 0  0 3 0 1 | 0  0  0  0  0 0 0 0  0  0  0 3  0 0 3 0 0 | 0 0 0  0  0 0 0 0  0  0  0 0 0  0  0  0  0 1 0 3  0  0 0 0 | * *  * * *  * * *  * 2 *  *  tet
0 0  0 2 1 1 | 0  0  0  0  0 0 0 0  0  0  0 1  2 0 2 0 1 | 0 0 0  0  0 0 0 0  0  0  0 0 0  0  0  0  0 0 1 1  0  2 0 0 | * *  * * *  * * *  * * 6  *  tet
0 0  0 1 2 1 | 0  0  0  0  0 0 0 0  0  0  0 0  1 1 1 1 2 | 0 0 0  0  0 0 0 0  0  0  0 0 0  0  0  0  0 0 0 0  1  1 1 1 | * *  * * *  * * *  * * * 12  tet