Tuza's conjecture for binary geometries

Abstract: Tuza (A conjecture, in Proceedings of the Colloquia Mathematica Societatis Janos Bolyai, 1981) conjectured that $\tau(G) \le 2\nu(G)$ for all graphs $G$, where $\tau(G)$ is the minimum size of an edge set whose removal makes $G$ triangle-free, and $\nu(G)$ is the maximum size of a collection of pairwise edge-disjoint triangles. Here, we generalise Tuza's conjecture to simple binary matroids that do not contain the Fano plane as a restriction. We prove that the geometric version of the conjecture holds for cographic matroids.