Dual-rank characterization of graphicness for binary matroids

Let $M$ be a $3$-connected binary matroid and let $S\subset E(M)$ satisfy $r^*_M(S)\ge 3$. A non-separating cocircuit is a cocircuit whose deletion leaves a connected matroid.

Graphicness characterization conjecture. The following assertions are equivalent:

1. $M$ is graphic.
2. Each element of $S$ avoids at most $r^*(M)-1$ non-separating cocircuits of $M$.
3. Each element of $S$ avoids no linearly dependent set of non-separating cocircuits of $M$.

The source presents this as a consequence of the main conjecture, whose proof was reduced to a computational verification. Thus the characterization remains conditional on that unresolved conjecture in the supplied paper.