Perfect-matching index four characterization of shortest cycle covers

Let $G$ be a bridgeless cubic graph. Write $scc(G)$ for the minimum total length of a cycle cover, and let $\chi'_{e}(G)$ denote the edge-chromatic number of $G$. Perfect-matching index four shortest-cycle-cover conjecture.

$$scc(G)=\frac{4}{3}\lvert E(G)\rvert\quad\Longleftrightarrow\quad \chi'_{e}(G)\leq 4.$$

The source proposes this equivalence and shows that it is implied by the shortest four-cycle-cover conjecture. It remains open.