Strong Ore-degree Chen–Lih–Wu decomposition conjecture

Let $k\geq3$, and let a $k$-decomposable graph be one admitting the $k$-decomposition defined in the source. Strong Ore-degree Chen–Lih–Wu conjecture. If $G$ is a $k$-colorable graph on $n$ vertices with $\Theta(G)\leq2k$, then $G$ has no equitable $k$-coloring if and only if $n$ is divisible by $k$ and there exists $W\subseteq V(G)$ such that $G[W]=K_{m,2k-m}$ for some odd $m$ and $G-W$ is $k$-decomposable. The source states that this stronger formulation is equivalent to the preceding conjecture in restricted settings.