The equitable total coloring conjecture

Let $G$ be a finite undirected graph without loops or multiple edges. An equitable total coloring is a total coloring in which the cardinalities of any two color classes differ by at most one. The equitable total chromatic number $\chi”_{=}(G)$ is the least number of colors in an equitable total coloring, and $\Delta(G)$ denotes the maximum degree of $G$.

Equitable total coloring conjecture. For every graph $G$,

$$\Delta(G)+1 \leq \chi”_{=}(G) \leq \Delta(G)+2.$$

The conjecture was proved for cubic graphs, so every cubic graph has an equitable total coloring with five colors. Its general status is not resolved in the supplied text.