The 6-regular three-2-factor conjecture

Let $G$ be a $6$-regular graph. A 2-factor is a spanning $2$-regular subgraph, and a component of a $2$-factor is one of its connected components. The 6-regular three-2-factor conjecture. Every $6$-regular graph is decomposable into three $2$-factors, where one of these $2$-factors has a component with at least $4$ vertices. This is identified as a sufficient condition for the $6$-regular case of the paper's main open problem. The corresponding assertion for $8$-regular graphs follows from a theorem on non-isomorphic $2$-factors, but the stated $6$-regular conjecture remains unresolved in the supplied text.