Tuza's conjecture for Fano-plane-free binary matroids

Let $M$ be a simple binary matroid. A triangle packing is a set of pairwise disjoint triangles of $M$, and a triangle hitting set is a set of elements meeting every triangle of $M$. Write $\nu(M)$ for the maximum size of a triangle packing and $\tau(M)$ for the minimum size of a triangle hitting set. Assume that $M$ does not contain a restriction isomorphic to the Fano plane. Tuza's conjecture for Fano-plane-free binary matroids.

$$\tau(M)\leq 2\nu(M).$$

This extends Tuza's graph conjecture to simple binary matroids excluding the Fano plane, which is a counterexample in the unrestricted binary-matroid setting. The paper proves the geometric version for cographic matroids, while the conjecture stated here remains open.