Meyer equitable chromatic number conjecture

Let $G$ be a connected graph. A proper equitable $t$-coloring is a proper vertex coloring whose color classes differ in size by at most one, and $\chi^{=}(G)$ is the least $t$ for which $G$ has a proper equitable $t$-coloring. Meyer's conjecture. If $G$ is neither a complete graph nor an odd cycle, then

$$\chi^{=}(G)\leq \Delta(G).$$

This is presented as a well-known conjecture on equitable colorings, attributed in the source to Meyer. Its resolution status is not specified in the supplied material.