Przybyło–Woźniak total 1-2 Conjecture

Let $G$ be a graph. A total coloring is a function $g:V\cup E\to\{1,2,\dots,k\}$, with total vertex weight $\sigma^t(v)=g(v)+\sum_{v\in e}g(e)$. Adjacent vertices are distinguished when their total weights differ, and $\chi^t_{\Sigma}(G)$ is the least $k$ for which every adjacent pair is distinguished. Total 1-2 Conjecture. For every graph $G$,

$$\chi^t_{\Sigma}(G)\leq 2.$$

The supplied text gives no resolution of this total neighbor-sum-distinguishing conjecture, so it remains open.