Cyclically 5-edge-connected odd 2-factor classification conjecture

Let $G$ be a cyclically $5$-edge-connected odd $2$-factored snark, meaning that $G$ is a snark, every cycle in every $2$-factor of $G$ is odd, and no edge cut of size at most four separates two components each containing a cycle. Let $P$ denote the Petersen graph and $J(t)$ the Flower snark. Cyclically 5-edge-connected classification conjecture. If $G$ is a cyclically $5$-edge-connected odd $2$-factored snark, then $G$ is either the Petersen graph or the Flower snark $J(t)$ for odd $t\ge 5$. This is a partial characterization problem for odd $2$-factored snarks; the source gives no resolution of this proposed classification.