Coefficient bounds conjecture for Ehrhart polynomials of connected matroids

Let $M$ be a connected matroid of rank $k$ on $n$ elements. Write $i(M,t)$ for its Ehrhart polynomial, and let $T_{k,n}$ and $U_{k,n}$ denote respectively the minimal and uniform matroids of rank $k$ on $n$ elements. For polynomials $P(t)$ and $Q(t)$, write $P(t)\preceq Q(t)$ when every coefficient of $P$ is at most the corresponding coefficient of $Q$. Coefficient bounds conjecture.

$$i(T_{k,n},t)\preceq i(M,t)\preceq i(U_{k,n},t).$$

This conjecture asks whether the minimal and uniform matroids give coefficientwise lower and upper bounds among connected matroids with fixed rank and cardinality. The containment of matroid polytopes in the corresponding hypersimplex gives the point-counting inequality, but coefficientwise bounds for the Ehrhart polynomials remain open in the supplied context.