Conjecture that nontrivial snarks of high perfect matching index are strong

Let $G$ be a nontrivial snark, and let its perfect matching index be the minimum number of perfect matchings covering all edges of $G$. An edge of a snark is suppressible if suppressing it preserves the relevant snark structure; the snark is strong when all its edges are suppressible.

Strong-snark conjecture. With the sole exception of the Petersen graph, every nontrivial snark with perfect matching index at least $5$ is strong; equivalently, all its edges are suppressible.

The conjecture is motivated by computations on thousands of nontrivial snarks, all of which were found to be strong. It does not extend to the paper's low-cyclic-connectivity examples, which have perfect matching index $5$ but contain non-removable edges, and its general status is open.