Brenti's log-concavity conjecture for Eulerian polynomials of labeled posets

Let $P$ be a poset with a labeling $\omega$, and let $\mathcal{L}(P,\omega)$ be its Jordan--Hölder set. Define the $(P,\omega)$-Eulerian polynomial by

$$W_{P,\omega}(t)=\sum_{\sigma\in\mathcal{L}(P,\omega)}t^{\operatorname{des}(\sigma)}.$$

Brenti's conjecture. The polynomial $W_{P,\omega}$ is log-concave with no internal zeroes.

The conjecture was posed by Brenti in 1989 after the Neggers--Stanley real-rootedness conjecture was disproved. The source states that it remains open for general labeled posets; it concerns unimodality and log-concavity of these descent-generating polynomials.