Strong Chen–Lih–Wu decomposition conjecture

Let $k\geq3$, and let a $k$-decomposition of a graph be a partition of its vertex set into induced subgraphs that are $k$-basic. Strong Chen–Lih–Wu conjecture. If $G$ is a $k$-colorable graph with $\Delta(G)=k$, then $G$ has no equitable $k$-coloring if and only if $k$ is odd and there exists $H\subseteq G$ such that $H=K_{k,k}$ and $G-H$ has a $k$-decomposition. This is presented as a strengthening intended to describe all non-equitably colorable graphs in the stated class.