The perfect-matching covering conjecture for claw-free bridgeless cubic graphs

Let $G$ be a claw-free bridgeless cubic graph, and let $k(G)$ be the smallest number of perfect matchings needed to cover the edge-set of $G$.

Claw-free Berge conjecture. For every claw-free bridgeless cubic graph $G$,

$$k(G)\leq 4.$$

This strengthens Berge's bound from five to four on claw-free bridgeless cubic graphs. The source presents it as a suspicion and uses it in connection with the $(5,2)$-even-subgraph-cover conjecture.