Faron–Postle Ore-degree conjecture for line-graph cliques

Let $G$ be a finite simple graph. For a non-empty subgraph $H$ of $G$, define its Ore-degree in $G$ by

$$\sigma_G(H):=\max_{xy\in E(H)}(\deg_G(x)+\deg_G(y)),$$

with $\sigma_G(H)=0$ when $H$ is empty. Suppose that $H$ is a bipartite subgraph of $G$ and that $E(H)$ forms a clique in $L(G)^2$. Faron–Postle conjecture. Then

$$|E(H)|\le \frac{1}{4}\sigma_G(H)^2.$$

This stronger Ore-degree statement implies the Faudree–Gyárfás–Schelp–Tuza strong clique index conjecture. The paper proves a substantial partial result with coefficient $\frac{620}{1987}$, but the conjectured coefficient $\frac14$ remains open.