Real-rootedness conjecture for Primitive Eulerian polynomials of simplicial arrangements

Let $\mathcal{A}$ be a simplicial arrangement, and let $P_{\mathcal{A}}(z)$ denote its Primitive Eulerian polynomial. Simplicial-arrangement real-rootedness conjecture. The polynomial $P_{\mathcal{A}}(z)$ is real-rooted. The source reports computational verification for several infinite and finite families, including $\mathcal{D}_{n,k}$ in the tested range and known crystallographic simplicial arrangements, but the assertion for all simplicial arrangements remains open.