Jackson–Sokal conjecture for characteristic polynomials of binary matroids

For a binary matroid $M$, let $\Lambda(M)$ be the minimum, over bases $B$ of its cocycle space, of the maximum size of a cocircuit in $B$. Let $C(M,t)$ denote its characteristic polynomial. Binary-matroid zero-free conjecture. There exists a constant $D$ such that for every loopless binary matroid $M$ and every complex number $t$ with

$$|t|\geq D\Lambda(M),$$

we have

$$C(M,t)\neq0.$$

This would extend the graph zero-free bound to binary matroids. The source then derives a corresponding flow-polynomial conjecture for bridgeless graphs; both assertions are presented without a resolution.