Kostochka–Kierstead Ore-degree Chen–Lih–Wu conjecture

For a graph $G$, define its Ore-degree by

$$\Theta(G):=\max\{d(x)+d(y):xy\in E(G)\}.$$

A proper coloring is equitable when its color classes differ in size by at most one. Kostochka–Kierstead's Ore-degree conjecture. Let $G$ be a connected graph with $\Theta(G)\leq2k$ and $k\geq3$. If $G$ is distinct from $K_{k+1}$ and from $K_{m,2k-m}$ for odd $m$, then $G$ has an equitable $k$-coloring. The source says this conjecture was proved for $k=3$ and relates it to a stronger equivalent formulation.