Cocircuit-rank conjecture for non-graphic binary matroids

Let $M$ be a $3$-connected non-graphic binary matroid. Define $Y(M)$ to be the set of elements of $M$ avoiding more than $r^*(M)-1$ non-separating cocircuits, and let $\widetilde{Y}(M):=E(M)-Y(M)$. Here $r^*_M$ denotes the rank function of the dual matroid.

Cocircuit-rank conjecture. One has

$$r^*_M(\widetilde{Y}(M))\le 2.$$

This conjecture generalizes the paper's principal theorem, which proves stronger conclusions for non-regular matroids and for regular matroids under specified minor hypotheses. The paper reports a theoretical reduction but says that the computational verification remains to be completed.