De Loera–Haws–Köppe conjecture on Ehrhart positivity of matroid basis polytopes

Let $M$ be a matroid and let $\mathscr{P}(M)$ denote its basis polytope. An integral polytope is Ehrhart positive if its Ehrhart polynomial has positive coefficients.

De Loera–Haws–Köppe conjecture. The basis polytope $\mathscr{P}(M)$ of every matroid $M$ is Ehrhart positive.

This was posed in 2007 and is one of the main open problems concerning Ehrhart positivity. The paper studies counterexamples to broader positivity expectations, while this matroid-basis-polytope conjecture remains open in the supplied source.