Descent formula conjecture for preorder-polytope h-star polynomials

Let $E$ be a totally ordered $n$-element set, let $\tau$ be a preorder on $E$, and let $\operatorname{des}(w)$ be the number of descents of a word $w\in E^n$. Define $\mathcal W_\tau$ to be the set of words $w\in E^n$ such that, for every order ideal $\mathcal I$ of $\tau$, the elements of $\mathcal I$ appear at least $|\mathcal I|$ times in $w$. Descent formula conjecture.

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

The conjecture extends a formula known for arbor polytopes to all preorder polytopes; its resolution is not specified in the source.