Chen–Lai–Wang equitable chromatic threshold conjecture

For a graph $G$, let $\Delta(G)$ denote its maximum degree and let $\chi_e^*(G)$ be the smallest integer $k$ such that $G$ is equitably $l$-colorable for every $l\geq k$. Chen–Lai–Wang’s conjecture. For any connected graph $G$, if it is different from a complete graph, a complete bipartite graph and an odd cycle, then

$$\chi_e^*(G)\leq\Delta(G).$$

The source records many partial results, including cases for bounded or sufficiently large maximum degree and several graph classes, but does not state that the conjecture is fully resolved.