The gentle arbor F-triangle conjecture

Let $t$ be an arbor, let $G_t$ be its gentle quiver-with-relations, and let $SC_t$ be the spherical simplicial complex polar dual to the simple polytope attached to $G_t$ by $\tau$-tilting theory. For a facet $C$ of $SC_t$, let the $F$-triangle of $(SC_t,C)$ denote the corresponding two-variable face enumerator. Let the transmuted $M$-triangle of $P_t$ be converted into an $F$-triangle by the formulas specified in the source.

The gentle arbor F-triangle conjecture. In $SC_t$, there exists a facet $C$ such that the $F$-triangle for $(SC_t,C)$ coincides with the $F$-triangle associated with the transmuted $M$-triangle of $P_t$.

This is a refinement of the support-$\tau$-tilting enumeration conjecture, relating the polar complex to the transmutation of the arbor poset's $M$-triangle. The source gives no proof or resolution.