Edwards–King edge-local conjecture for graph coloring

Let $G$ be a finite simple graph. For each vertex $v$, let $\widetilde N(v)=\{v\}\cup N(v)$, let $\gamma_\ell(v)=\gamma(G[\widetilde N(v)])$, and let

$$\gamma(G)=\left\lceil\frac{\Delta(G)+1+\omega(G)}{2}\right\rceil.$$

Edwards–King conjecture. For any graph $G$,

$$\chi(G)\leq\max_{uv\in E(G)}\left\lceil\frac{\gamma_\ell(u)+\gamma_\ell(v)}{2}\right\rceil.$$

This is a stronger edge-local form of the local Reed conjecture. The source says that the result holds in the fractional setting and for quasi-line graphs, while its validity for arbitrary graphs remains unresolved.