I have a question on Total conjecture in graph theory. Total conjecture states that $$\chi^"(G)\leq \Delta +2,$$ where $\Delta$ is the maximum degree of the graph and $\chi^"(G)$ denotes the total coloring (minimum number of colors for coloring graph such that no adjacent edges and no edge and its endpoints are assigned the same color.) number.

Question: Does total conjecture has been proved for complete graphs?