The $\Delta$-Equitable Coloring Conjecture

Let $G$ be a connected finite simple graph, and let $\Delta(G)$ denote its maximum vertex degree. Write $K_m$ for the complete graph on $m$ vertices, $C_{2m+1}$ for an odd cycle, and $K_{2m+1,2m+1}$ for the complete bipartite graph with parts of size $2m+1$.

The $\Delta$-ECC. $G$ is equitably $\Delta(G)$-colorable if it is different from $K_m$, $C_{2m+1}$, and $K_{2m+1,2m+1}$.

The conjecture is a list analogue of Brooks's theorem for equitable coloring. It has been proved for interval graphs, trees, outerplanar graphs, subcubic graphs, and several other graph classes.