The linear bound for 1-strongly vertex-distinguishing total coloring

Let $G$ be a graph with no isolated edges, let $\Delta(G)$ be its maximum degree, and let $\chi_{1\text{-vsdt}}(G)$ denote its 1-strongly vertex-distinguishing total chromatic number.

Linear vertex-distinguishing total coloring conjecture. For some positive constant $c$,

$$\chi_{1\text{-vsdt}}(G)\le 2\Delta(G)+c.$$

The conjecture is motivated by Hatami's result that, for sufficiently large maximum degree, the corresponding quantity is at most $2\Delta(G)+300$. Whether a single positive constant works for every graph with no isolated edges is left open in the source.