I have a question on the Total Coloring Conjecture in graph theory. This 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: Has the Total Coloring Conjecture been proved for complete graphs?