The near-bipartite stability conjecture for low fractional triangle-packing density

Let $G$ be a graph on $n$ vertices, let $\overline{G}$ denote its complement, and define

$$\eta(G):=\frac{\nu^*(G)+\nu^*(\overline{G})}{n(n-1)},$$

where $\nu^*(G)$ is the fractional triangle-packing number of $G$. Near-bipartite stability conjecture. If $n\geq 26$ and $\eta(G)\leq 1/4$, then either $G$ or $\overline{G}$ can be made bipartite by removing at most $n/8$ edges.

The conjecture identifies the apparent obstruction to having $\eta(G)>1/4$: a graph or its complement may be close to bipartite. The paper notes that proving it would require bridging computational evidence for small $n$ and stability arguments for larger $n$, and describes the problem as harder for general graphs than for triangle-free graphs.