Kaiser's lower-bound conjecture for three perfect matchings

Let $G$ be a bridgeless cubic graph, and let $m_3(G)$ be the ratio of the maximum number of edges covered by the union of three perfect matchings to $|E(G)|$.

Kaiser's conjecture. For every bridgeless cubic graph $G$,

$$m_3(G)\geq \frac45.$$

Kaiser et al. proved the weaker bound $m_3(G)\geq 27/35\approx 0.77$ and conjectured that $4/5$ is the best possible lower bound, attained by the Petersen graph.