Weighted Berge–Fulkerson matching conjecture for bridgeless cubic graphs

Let $G$ be a weighted bridgeless cubic graph, meaning that $G$ has no bridges, every vertex has degree $3$, and its edges have non-negative weights. Let $w(G)$ be the total edge weight and let $M$ be a perfect matching, a set of pairwise disjoint edges meeting every vertex exactly once.

Weighted perfect-matching conjecture. There exists a perfect matching $M$ such that

$$w(M)\geq \frac{w(G)}{3}.$$

This is presented as a weighted consequence of the Berge–Fulkerson conjecture and would imply the relevant case of the paper's triangle-free cubic bisection conjecture. Its general validity remains open.