The {(hhH)²} face (in the naming: B for barrel) has corner angles of h = 108° and H = 144°. Note that h is the same corner angle as the pentagon, thence that non-regular hexagon reuses 4 out of 5 vertices of such a virtually inscribed pentagon. The {(rR)²} face (in the naming: R for rhomb) is just the union of 2 coplanar triangles, i.e. r = 60° and R = 120°. Esp. HH = BD = B'D' = f and RR = CD' = C'D = x.

In 2025 a discord member, calling himself "puffer fish" came up with this convex, just rotoreflectional, all unit edged polyhedron. He also provided the following derivation pics:

©

Incidence matrix

1 * * * * * * * * * | 3 0 0 0 0 0 0 0 0 0 0 0 0 0 | 3 0 0 0 0 0  A  [h3]
* 3 * * * * * * * * | 1 1 1 0 0 0 0 0 0 0 0 0 0 0 | 2 1 0 0 0 0  B  [4,h,H]
* * 3 * * * * * * * | 0 1 0 1 1 0 0 0 0 0 0 0 0 0 | 1 1 1 0 0 0  C  [4,h,R]
* * * 3 * * * * * * | 0 0 1 0 0 1 1 0 0 0 0 0 0 0 | 1 1 0 1 0 0  D  [4,H,R]
* * * * 3 * * * * * | 0 0 0 1 0 1 0 1 1 0 0 0 0 0 | 1 0 1 1 1 0  E  [4,r,h,r]
* * * * * 3 * * * * | 0 0 0 0 1 0 1 0 0 1 1 0 0 0 | 0 1 1 1 0 1  E' [4,r,h,r]
* * * * * * 3 * * * | 0 0 0 0 0 0 0 1 0 1 0 1 0 0 | 0 0 1 0 1 1  D' [4,H,R]
* * * * * * * 3 * * | 0 0 0 0 0 0 0 0 1 0 1 0 1 0 | 0 0 0 1 1 1  C' [4,h,R]
* * * * * * * * 3 * | 0 0 0 0 0 0 0 0 0 0 0 1 1 1 | 0 0 0 0 1 2  B' [4,h,H]
* * * * * * * * * 1 | 0 0 0 0 0 0 0 0 0 0 0 0 0 3 | 0 0 0 0 0 3  A' [h3]
--------------------+-----------------------------+------------
1 1 0 0 0 0 0 0 0 0 | 3 * * * * * * * * * * * * * | 2 0 0 0 0 0
0 1 1 0 0 0 0 0 0 0 | * 3 * * * * * * * * * * * * | 1 1 0 0 0 0
0 1 0 1 0 0 0 0 0 0 | * * 3 * * * * * * * * * * * | 1 1 0 0 0 0
0 0 1 0 1 0 0 0 0 0 | * * * 3 * * * * * * * * * * | 1 0 1 0 0 0
0 0 1 0 0 1 0 0 0 0 | * * * * 3 * * * * * * * * * | 0 1 1 0 0 0
0 0 0 1 1 0 0 0 0 0 | * * * * * 3 * * * * * * * * | 1 0 0 1 0 0
0 0 0 1 0 1 0 0 0 0 | * * * * * * 3 * * * * * * * | 0 1 0 1 0 0
0 0 0 0 1 0 1 0 0 0 | * * * * * * * 3 * * * * * * | 0 0 1 0 1 0
0 0 0 0 1 0 0 1 0 0 | * * * * * * * * 3 * * * * * | 0 0 1 0 0 1
0 0 0 0 0 1 1 0 0 0 | * * * * * * * * * 3 * * * * | 0 0 0 1 1 0
0 0 0 0 0 1 0 1 0 0 | * * * * * * * * * * 3 * * * | 0 0 0 1 0 1
0 0 0 0 0 0 1 0 1 0 | * * * * * * * * * * * 3 * * | 0 0 0 0 1 1
0 0 0 0 0 0 0 1 1 0 | * * * * * * * * * * * * 3 * | 0 0 0 0 1 1
0 0 0 0 0 0 0 0 1 1 | * * * * * * * * * * * * * 3 | 0 0 0 0 0 2
--------------------+-----------------------------+------------
1 2 1 1 1 0 0 0 0 0 | 2 1 1 1 0 1 0 0 0 0 0 0 0 0 | 3 * * * * *  {(hhH)2}
0 1 1 1 0 1 0 0 0 0 | 0 1 1 0 1 0 1 0 0 0 0 0 0 0 | * 3 * * * *  {4}
0 0 1 0 1 1 1 0 0 0 | 0 0 0 1 1 0 0 1 1 0 0 0 0 0 | * * 3 * * *  {(rR)2}
0 0 0 1 1 1 0 1 0 0 | 0 0 0 0 0 1 1 0 0 1 1 0 0 0 | * * * 3 * *  {(rR)2}
0 0 0 0 1 0 1 1 1 0 | 0 0 0 0 0 0 0 1 0 1 0 1 1 0 | * * * * 3 *  {4}
0 0 0 0 0 1 1 1 2 1 | 0 0 0 0 0 0 0 0 1 0 1 1 1 2 | * * * * * 3  {(hhH)2}

or
2 * * * * | 3 0 0 0 0 0 0 | 3 0 0  A + A'  [h3]
* 6 * * * | 1 1 1 0 0 0 0 | 2 1 0  B + B'  [4,h,H]
* * 6 * * | 0 1 0 1 1 0 0 | 1 1 1  C + C'  [4,h,R]
* * * 6 * | 0 0 1 0 0 1 1 | 1 1 1  D + D'  [4,H,R]
* * * * 6 | 0 0 0 1 1 1 1 | 1 1 2  E + E'  [4,r,h,r]
----------+---------------+------
1 1 0 0 0 | 6 * * * * * * | 2 0 0
0 1 1 0 0 | * 6 * * * * * | 1 1 0
0 1 0 1 0 | * * 6 * * * * | 1 1 0
0 0 1 0 1 | * * * 6 * * * | 1 0 1  CE + C'E'
0 0 1 0 1 | * * * * 6 * * | 0 1 1  CE' + C'E
0 0 0 1 1 | * * * * * 6 * | 1 0 1  DE + D'E'
0 0 0 1 1 | * * * * * * 6 | 0 1 1  DE' + D'E
----------+---------------+------
1 2 1 1 1 | 2 1 1 1 0 1 0 | 6 * *  {(hhH)2}
0 1 1 1 1 | 0 1 1 0 1 0 1 | * 6 *  {4}
0 0 1 1 2 | 0 0 0 1 1 1 1 | * * 6  {(rR)2}