A Sharp Basis-Mass Inequality for Binary Matroids without a Fano Minor

Let $M$ be a binary matroid of rank $r$ with no $F_7$-minor, and let $B_M^\star$ denote the maximum of its basis-generating polynomial over nonnegative weights of total mass $r$. We prove the sharp inequality $B_M^\star\leq 2^r/(r+1)$, with equality holding if and only if the nonloop simplification