The matching characterization of the proper Z4 x Z2-coloring conjecture

Let $G$ be a bridgeless cubic graph, let $F$ be a $2$-factor of $G$, and let $M$ be a matching in $F$. Put $H=G-M$. An $F$-matching is the matching notion defined in the paper, and an $F$-complement is called $3$-even when it has the property defined there. Matching characterization conjecture. Every bridgeless cubic graph $G$ has a $2$-factor $F$ and a matching $M$ in $F$ such that $H=G-M$ has an $F$-matching whose $F$-complement is $3$-even. This is presented as an equivalent reformulation of the proper abelian coloring conjecture for the exceptional groups, so its resolution would settle the existence of proper $\mathbb{Z}_{4}\times\mathbb{Z}_{2}$-colorings for all bridgeless cubic graphs.