The critical-snark Fulkerson-cover conjecture

Let $G$ be a critical snark, let a hexagon be a $6$-cycle, and let a hexagon be double-core when it is a double-core hexagon for a $3$-array of perfect matchings. An optimal $3$-array is a $3$-array leaving the minimum possible number of uncovered edges. Critical-snark Fulkerson-cover conjecture. Every hexagon in $G$ is double-core. In particular, every optimal $3$-array of perfect matchings extends to a Fulkerson cover. The claim is motivated by the fact that a double-core hexagon yields a Fulkerson cover; its general validity is left open.