Athanasiadis–Kalampogia-Evangelinou's real-rootedness conjecture for Bergman-complex h-polynomials

Let $\mathrm{M}$ be a matroid, let $\mathcal{L}(\mathrm{M})$ be its lattice of flats, and let $\Delta(\widehat{\mathcal{L}}(\mathrm{M}))$ be the order complex of its proper nonempty flats, also called the Bergman complex. Athanasiadis–Kalampogia-Evangelinou's conjecture. The $h$-polynomial of $\Delta(\widehat{\mathcal{L}}(\mathrm{M}))$ has only nonpositive real roots. This is the $h$-polynomial reformulation of the conjectured real-rootedness of the chain polynomial of a geometric lattice; the source gives no resolution.