Neggers-Stanley conjecture for W-polynomials of labeled posets

Let $(P,\omega)$ be a labeled poset. Its $W$-polynomial is defined by

$$\sum_{n \geq 0}\Omega(P,\omega;n+1)t^n = \frac{W(P,\omega;t)}{(1-t)^{p+1}}.$$

Equivalently, if $\mathcal{L}(P,\omega)$ is the set of permutations obtained by applying $\omega$ to linear extensions of $P$, and $\operatorname{des}(\pi)$ denotes the number of descents of a permutation $\pi$, then

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

Neggers-Stanley conjecture. For any labeled poset $(P,\omega)$, the polynomial $W(P,\omega;t)$ has only real zeros.

The conjecture generalizes Neggers's 1978 conjecture for naturally labeled posets and Stanley's 1986 conjecture for arbitrary labelings. It is known in several special cases, but the supplied source gives no resolution of the general statement.