Degree-two 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-two lemma are equivalent. Degree-two characterization conjecture.

The preceding theorem proves the equivalence, with the additional condition (a$'$), for graphic and cographic matroids, while the example $F_7^*$ shows that (a$'$) need not hold for every simple and cosimple connected binary matroid. The conjecture asks whether (a), (b), and (c) alone are nevertheless equivalent in this broader class.