Degree-normalized ultra log-concavity for graphic and general matroids

Determine whether every graphic matroid, and more generally every matroid, has a degree-normalized ultra log-concave Ehrhart h*-polynomial.

Background

The paper proves that every matroid of rank two or corank two has a positive Ehrhart h*-coefficient sequence that is ultra log-concave of order equal to the degree of the h*-polynomial. This result extends beyond real-rootedness, since the paper also exhibits rank-two examples whose h*-polynomials are not real-rooted.

The concluding question asks whether the same degree-normalized ultra log-concavity property holds for all graphic matroids and, more broadly, for arbitrary matroids. The authors identify nonnegative coefficient expansions together with compatible upper and lower coefficient bounds as one possible approach in higher rank, but do not resolve the question.