Real-rootedness and Eulerian interlacing conjecture for geometric lattices

Let ${\mathcal L}$ be a finite geometric lattice of rank $n$, and write

$$h_{\mathcal L}(x)=\sum_{i=0}^{n-1}a_i x^i.$$

Let $A_n(x)$ be the Eulerian polynomial of index $n$. A polynomial is interlaced by another polynomial when their roots satisfy the interlacing relation used in the paper.

Real-rootedness and Eulerian interlacing conjecture. The polynomial $h_{\mathcal L}(x)$ has only real roots and is interlaced by the Eulerian polynomial $A_n(x)$ for every geometric lattice ${\mathcal L}$ of rank $n$.

The conjecture is stronger than the open unimodality question for $h_{\mathcal L}(x)$ and had been verified computationally for all geometric lattices with at most nine atoms. No general proof or disproof is given in the supplied text.