The arbor-polytope word-enumerator conjecture

Let $\tau$ be an arbor of size $n$, let $\mathcal Q_\tau$ be its associated arbor polytope, and let $\mathcal D(v)$ denote the union of the blocks at the descendants of a vertex $v$, including $v$ itself. Let $\mathcal W_\tau$ be the set of words $w\in[n]^n$ such that, for every vertex $v$ of $\tau$, the elements of $\mathcal D(v)$ appear in $w$ at least $|\mathcal D(v)|$ times in total. Writing ${\rm des}(w)$ for the number of descents of $w$, the arbor-polytope word-enumerator conjecture.

$$h^\ast(\mathcal Q_\tau,t)=\sum_{w\in\mathcal W_\tau}t^{n-1-{\rm des}(w)}.$$

This is presented as a conjectural generalization of the preceding theorem and is supported in the source by extensive computer experimentation; no proof or resolution is supplied.