Vignesh et al.'s total coloring conjecture for line graphs of complete graphs

Let $K_n$ be the complete graph on $n$ vertices, let $L(K_n)$ be its line graph, and let $\chi”$ denote total chromatic number. Vignesh et al.'s conjecture. For every complete graph $K_n$,

$$\chi”(L(K_n))=2n-3.$$

The claim extends the directly verified cases $n\leq4$ and predicts that every line graph of a complete graph is type-I. The supplied text gives no resolution.