Kierstead-Kostochka-Mydlarz-Szemerédi algorithmic Ore-type conjecture

Let an equitable $k$-coloring be a proper $k$-coloring whose color classes have sizes differing by at most one, and let $d(x)$ denote the degree of a vertex $x$. Kierstead-Kostochka-Mydlarz-Szemerédi conjecture. If $G$ is a graph with $d(x)+d(y)<2k$ for all $xy∈E(G)$, then there exists a polynomial-time algorithm to determine an equitable $k$-coloring of $G$.

This is the algorithmic counterpart of the Ore-type equitable-coloring conjectures. The source says it holds when $k$ is at least a fixed positive proportion of the order of $G$, while smaller values are not covered by the paper.