F-triangle realization for unary-binary arbors

Construct, for every unary-binary arbor t, a maximal face C of the polar simplicial complex SC_t of the simple polytope associated with the gentle quiver-with-relations G_t such that the F-triangle of (SC_t,C) coincides with the F-triangle obtained from the transmuted M-triangle of P_t by the stated conversion formulas.

Background

The gentle quiver-with-relations G_t has an associated simple polytope whose polar dual is the spherical simplicial complex SC_t. The paper defines transmutation as a birational involution on M-triangles and seeks geometric realizations of the resulting polynomial through F-triangles. The conjecture asks for a distinguished maximal face of SC_t producing exactly the F-triangle predicted from P_t. It was checked computationally for arbors of size at most 8.