Deloera–Haws–Köppe conjecture for matroid polytopes

Let $M$ be a matroid on $[d]:=\{1,\ldots,d\}$, with bases $\mathcal{B}$, and let

$$P(M):=\operatorname{conv}\left(\sum_{i\in B}{\boldsymbol{e}}_i:B\in\mathcal{B}\right)\subset\mathbb{R}^d$$

be its matroid polytope. A lattice polytope is Ehrhart positive if all coefficients of its Ehrhart polynomial are positive, and Ehrhart unimodal if its Ehrhart $h^\ast$-polynomial has a unimodal coefficient sequence.

Deloera–Haws–Köppe conjecture. For any matroid $M$, the matroid polytope $P(M)$ is both Ehrhart positive and Ehrhart unimodal.

Both aspects of this conjecture were described as elusive, with only partial results known; the conjecture concerns a broad family beyond the zonotopes already known to have both properties.