Interlacing conjecture for Kazhdan–Lusztig polynomials of matroids

Let $M$ be a matroid of positive rank, let $e$ be an element of its ground set, and let $M/e$ be the contraction of $M$ at $e$. Define

$$Q_M(t):=t^{\operatorname{rk} M-1}P_M(-t^{-2}),$$

where $P_M(t)$ is the Kazhdan–Lusztig polynomial of $M$. Say that a polynomial $f(t)$ interlaces a polynomial $g(t)$ when both are real rooted and their roots alternate, starting with the smallest root of $f(t)$. A matroid is non-degenerate if its Kazhdan–Lusztig polynomial has the maximal degree allowed by its rank.

Interlacing conjecture. If $M$ and $M/e$ are both non-degenerate, then $Q_M(t)$ interlaces $Q_{M/e}(t)$.

If the real-rootedness conjecture holds, the roots of $Q_M(t)$ are real and symmetric about the origin. In odd rank the claim is equivalent to interlacing of $P_M(t)$ and $P_{M/e}(t)$; in even rank it is equivalent to interlacing of $tP_{M/e}(t)$ and $P_M(t)$.