The four matroid polynomial conjectures

Let $\mathsf{M}$ be a matroid, with Kazhdan–Lusztig polynomial $P_{\mathsf{M}}(x)$, $Z$-polynomial $Z_{\mathsf{M}}(x)$, inverse Kazhdan–Lusztig polynomial $Q_{\mathsf{M}}(x)$, and inverse $Z$-polynomial $Y_{\mathsf{M}}(x)$.

The four matroid polynomial conjectures. For every matroid $\mathsf{M}$, the polynomial $P_{\mathsf{M}}(x)$ is real-rooted, the polynomial $Z_{\mathsf{M}}(x)$ is real-rooted, the polynomial $Q_{\mathsf{M}}(x)$ has log-concave coefficients, and the polynomial $Y_{\mathsf{M}}(x)$ has log-concave coefficients.

The first, third, and fourth claims are supported primarily by computations for special families of matroids. For the second, $Z$-polynomials are known to be $\gamma$-positive, which is weaker than real-rootedness but stronger than unimodality; all four claims remain open here.