The $(k)$-total coloring conjecture

Let $k\geq2$ be a fixed positive integer and let $G$ be a graph. $(k)$-total coloring conjecture.

$$\chi”(G)\leq\Delta(G)+k.$$

This is presented as a weaker version of the total coloring conjecture obtained by relaxing the upper bound. The case $k=2$ is the original total coloring conjecture, while larger $k$ give weaker assertions; the supplied text does not indicate a resolution of this general formulation.