Meyer's conjecture on the equitable chromatic number

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

Meyer's conjecture.

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

with the exception that $G$ is a complete graph or an odd cycle.

This is one of the two stated open conjectures concerning equitable colouring parameters.