The strong-snark conjecture for nontrivial cubic graphs

Let $G$ be a nontrivial cubic graph, meaning here a cubic graph with cyclic connectivity at least $4$ and girth at least $5$. Its perfect matching index is the minimum number of perfect matchings whose union contains every edge of $G$. A strong snark is a cubic graph such that deleting any edge yields a graph homeomorphic to a cubic graph admitting no proper $3$-edge-colouring.

Strong-snark conjecture. If the perfect matching index of $G$ is greater than $4$ and $G$ is not the Petersen graph, then $G$ is a strong snark.

The conjecture was proposed in the cited work and addresses the structure of nontrivial cubic graphs requiring at least five perfect matchings. Its resolution is not stated in the supplied text.