Cycle-double-cover characterization of edge-colouring index four

Let $G$ be a bridgeless cubic graph. A cycle double cover is a collection of cycles covering every edge exactly twice, and a $2$-factor is a spanning $2$-regular subgraph. Let $\chi'_{e}(G)$ denote the edge-chromatic number of $G$. Cycle-double-cover characterization conjecture.

$$\chi'_{e}(G)>4\quad\Longleftrightarrow\quad\text{every cycle double cover of }G\text{ does not contain a }2\text{-factor of }G.$$

The source states this as an equivalent formulation of its perfect-matching index four and shortest-cycle-cover conjecture. It remains open.