Weighted triangle-free subcubic bisection conjecture

Let $G=(V,E,w)$ be a weighted triangle-free subcubic graph, meaning that $G$ has no triangle and maximum degree at most $3$, with non-negative edge weights and total edge weight $w(G)$. A bisection is a cut whose two parts differ in size by at most $1$.

Triangle-free subcubic bisection conjecture. Every weighted triangle-free subcubic graph other than $K_{1,3}$ has a bisection of weight at least

$$\frac{11}{15}w(G).$$

The paper proves a lower bound of $\frac{613}{855}w(G)$ for graphs other than $K_{1,3}$ and notes that the Petersen graph may make $\frac{11}{15}$ optimal; the conjectured improvement remains open.