Mkrtchyan's $S_{12}$-Conjecture

Let $G$ be a cubic graph admitting a perfect matching, and let $S_{12}$ denote the multigraph introduced in the source. For graphs $H$ and $G$, write $H\prec G$ when $G$ admits an $H$-colouring.

$S_{12}$-Conjecture. For each cubic graph $G$ admitting a perfect matching, $S_{12}\prec G$.

This is one of two conjectures proposed by Mkrtchyan for cubic graphs without a bridgelessness assumption. The supplied source gives no resolution status.