Degree-five characterization conjecture for simple or cosimple connected binary matroids

Let $M$ be a connected binary matroid that is simple or cosimple. The statements (a), (b), and (c) in the degree-five lemma are equivalent. Degree-five characterization conjecture.

The lemma gives only the implications from (a) to (b) to (c) for binary matroids, so this conjecture proposes equivalence for connected matroids that are simple or cosimple. An explicit characterization of matroids $M$ with the relevant invariant satisfying $\b5(M)\leq 5$ is presented as an alternative motivating problem, and the conjectured equivalence remains open in the supplied text.