The matching-deletability conjecture for 3-edge-connected graphs

Let $G$ be a $3$-edge-connected graph. A matching is a set of pairwise vertex-disjoint edges, and a 3-edge-cut is an edge cut of size three. A matching is deletable when all its edges can become deletable in one orientation, as defined in the paper. Matching-deletability conjecture. If $M$ is a matching of $G$ intersecting each $3$-edge-cut of $G$ in at most one edge, then $M$ is deletable. This proposed generalization is presented as a possible route toward improving the Frank-number bound; its status is open in the supplied text.