Maximum-weight bisection bound for graphs of bounded maximum degree

Let $G=(V,E,w)$ be an edge-weighted graph with non-negative edge weights, and let $w(G)$ denote the total edge weight. A bisection is a cut whose two parts differ in size by at most $1$. Let $k$ be a positive integer and suppose that

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

Bounded-degree bisection conjecture. If $k$ is odd, then $G$ has a bisection of weight at least

$$\frac{k+1}{2k}w(G),$$

and if $k$ is even, then $G$ has a bisection of weight at least

$$\frac{k+2}{2(k+1)}w(G).$$

The corresponding bounds are known for maximum cuts and are tight there for complete graphs; the conjecture asks whether they also hold for maximum-weight bisections. The paper notes that the conjecture holds for weighted subcubic graphs and almost all graphs, while the general case remains open.