Strong equitable Brooks conjecture

Let $G$ be a graph. A graph is $k$-equitable if $|G|=ks$, $\chi(G)\le k$, and every proper $k$-coloring of $G$ has $s$ vertices in each color class. Write $H_1+H_2$ for the disjoint union of graphs $H_1$ and $H_2$. Strong equitable Brooks conjecture. Every graph $G$ with $\chi(G), \Delta(G)\le k$ has no equitable $k$-coloring if and only if $k$ is odd and $G=H+K_{k,k}$ for some $k$-equitable graph $H$. This is presented as a strengthening of the Chen–Lih–Wu conjecture; the cited results leave cases with $k\ge5$ and at least five color classes unresolved.