The vertex-distinguishing total coloring bound

Let $G$ be a graph on $n$ vertices with no isolated edges, and let $\chi_{r\text{-vsdt}}(G)$ denote its $r$-strongly vertex-distinguishing total chromatic number.

Vertex-distinguishing total coloring conjecture.

$$\chi_{r\text{-vsdt}}(G)\le n+\lceil\log_{2}n\rceil+1$$

\nand equality holds if $n=2^{k}-2$.

This conjecture proposes a general upper bound based on the number of vertices; the equality condition is motivated by the known values for complete graphs. Its resolution is not supplied in the source.