Fleischner's conjecture on compatible cycle decompositions

Let $G$ be an essentially $6$-edge-connected $4$-regular graph with a transition system $T$. A compatible cycle decomposition is a cycle decomposition compatible with the transitions prescribed by $T$.

Fleischner's conjecture. The pair $(G,T)$ has no compatible cycle decomposition if and only if $(G,T)$ is the bad loop or the bad $K_5$.

The source lists this conjecture among claims for which finitely many counterexamples had already been found by computer search, and its construction is presented as part of the counterexample context.