Aldred–Labbate–Robertson–Seymour conjecture on cyclically 5-edge-connected odd 2-factored snarks

Let $G$ be a cyclically $5$-edge-connected odd $2$-factored snark, where a snark is a bridgeless cubic graph of chromatic index four and odd 2-factored means that every cycle in every $2$-factor is odd. Let $P$ be the Petersen graph and $J(t)$ the Flower snark. Aldred–Labbate–Robertson–Seymour's conjecture. The graph $G$ is either the Petersen graph or the Flower snark $J(t)$ for odd $t\geq5$. This is a proposed partial characterization after earlier constructions produced counterexamples to the broader odd $2$-factored snark conjecture.