The constant-additive neighbour sum distinguishing total colouring conjecture

Let $G$ be a graph with maximum degree $\Delta$, and let $\chi”_{\sum}(G)$ denote the least number of colours in a proper total colouring whose weighted degrees distinguish adjacent vertices. Constant-additive conjecture. There exists a constant $C$ such that

$$\chi”_{\sum}(G)\leq \Delta(G)+C$$

for every graph $G$. This is explicitly described as a weaker version of the Pilśniak–Woźniak conjecture; the source proves only an asymptotic bound, so this weaker constant-additive statement remains open there.