Havet and Yu's $(p,1)$-total labelling conjecture

Let $G$ be a finite, simple, undirected graph, let $p$ be a positive integer, and let $\lambda^T_p(G)$ denote the minimum $k$ for which $G$ has a $(p,1)$-total $k$-labelling. Havet and Yu's conjecture.

$$\lambda^T_p(G)\leq \min\\{\Delta(G)+2p-1,2\Delta(G)+p-1\\}.$$

For $p=1$, this is equivalent to the Total Coloring Conjecture, namely $\chi”(G)\leq \Delta(G)+2$. The source presents the conjecture as open, while noting that substantial special cases are known.