Chen–Lih–Wu conjecture on the equitable chromatic threshold

Let $G$ be a connected graph, and let $\chi_{\equiv}(G)$ denote its equitable chromatic threshold and $\Delta(G)$ its maximum degree.

Chen–Lih–Wu conjecture.

$$\chi_{\equiv}(G)\leq \Delta(G),$$

with the exception that $G$ is a complete graph, an odd cycle, or a complete bipartite graph $K_{2m+1,2m+1}$.

The supplied source does not state whether this conjecture has been resolved, so it is recorded as open.