Zhang et al.'s AVD total coloring conjecture

Let $G$ be a finite, simple, undirected graph. For each vertex $u$, let $N_G(u)$ be its set of neighbours, let $\partial_G(u)$ be the set of edges incident with $u$, and let $\partial_G^\star(u)=\{u\}\cup\partial_G(u)$. Write $\Delta(G)$ for the maximum degree of $G$. An adjacent vertex distinguishing (AVD) total coloring is a proper total coloring $\phi:V(G)\cup E(G)\to C$ such that $\phi(\partial_G^\star(u))\ne\phi(\partial_G^\star(v))$ for every edge $uv\in E(G)$. AVD Total Coloring Conjecture. Every graph $G$ has an AVD total coloring using at most $\Delta(G)+3$ colors.

The conjecture is a strengthening of the total coloring problem, requiring adjacent vertices to have distinct sets of colors on their incident vertices and edges. The source attributes it to Zhang et al.; the supplied text does not establish whether it is resolved in full.