Ultra-log-concavity of weak-order Eulerian polynomials

Let $\pi\in\mathfrak{S}_n$. The polynomial

$$Q_W^\pi(t)=\sum_{\sigma\in[\operatorname{id},\pi]_W}t^{\operatorname{des}(\sigma)}$$

is the weak-order Eulerian polynomial of the interval below $\pi$.

Ultra-log-concavity conjecture. For every $\pi\in\mathfrak{S}_n$, $Q_W^\pi$ is ultra-log-concave.

This is a special case of Brenti's conjecture, rephrased in the language of the weak order. The claim is motivated by ultra-log-concavity results for naturally labeled width-two posets, although the corresponding polynomials for general permutation posets need not be real-rooted.