Brenti's real-rootedness conjecture for labeled-poset Eulerian polynomials

Let $P$ be a finite poset of cardinality $p$, and let $\omega$ be a labeling of $P$, namely a bijection from $P$ to $\{1,2,\ldots,p\}$. Let $\mathcal{L}(P,\omega)$ be the Jordan–Hölder set of permutations obtained from linear extensions of $P$, and define the labeled-poset Eulerian polynomial by

$$W(P,\omega;t)=\sum_{\pi\in\mathcal{L}(P,\omega)}t^{\operatorname{des}(\pi)}.$$

Brenti's conjecture. For any labeled poset $(P,\omega)$, the polynomial $W(P,\omega;t)$ has only real zeros as a polynomial in $t$. This is the Poset Conjecture, also called the Neggers–Stanley Conjecture. The conjecture is presented in the source as a result attributed to Brenti, but the supplied material gives no resolution, so its database status remains open.