Mkrtchyan–Hakobyan $S_{12}$-Conjecture

Let $G$ be a finite graph, and for graphs $G$ and $H$ write $H\prec G$ when there is an $H$-coloring of $G$: a mapping $f\colon E(G)\to E(H)$ that maps adjacent edges to distinct edges and satisfies $f(\partial_G(v))=\partial_H(u)$ for some $u\in V(H)$ at every vertex $v\in V(G)$. A graph is cubic if every vertex has degree three, and a perfect matching is a set of pairwise nonadjacent edges meeting every vertex exactly once. Let $S_{12}$ be obtained from $S_{10}$ by replacing its central vertex with a triangle.

$S_{12}$-Conjecture. If $G$ is a cubic graph with a perfect matching, then

$$S_{12}\prec G.$$

The conjecture was proposed by Mkrtchyan and Hakobyan; the supplied source announces a counterexample to it, so the assertion is refuted.