Infinite cyclically 6-edge-connected counterexamples to non-conflicting flows

Let $G$ be a cubic graph and let $F$ be a perfect matching; write $\overline{F}$ for the complementary $2$-factor. A non-conflicting nowhere-zero $Z_2\times Z_2$-flow with respect to $\overline{F}$ is a nowhere-zero flow on the contraction associated with $\overline{F}$ satisfying the paper's non-conflict condition. The cyclically 6-edge-connected counterexample conjecture. There exist infinitely many cyclically 6-edge-connected cubic graphs $G$ having no such flow with respect to $\overline{F}$ for any perfect matching $F$. This is proposed as a negative answer to a uniqueness question about the Petersen graph and remains open.