The 1-2 Conjecture for total multiset-labellings

Let $G$ be a graph. A total $k$-labelling is a map $\ell:V(G)\cup E(G)\to\{1,\dots,k\}$. For each vertex $u$, let $\mu_\ell(u)$ be the multiset of labels incident to $u$, including the label of $u$. The labelling is multiset-proper when adjacent vertices have distinct incident multisets, and $\chi^t_{\rm M}(G)$ is the least $k$ for which such a total labelling exists. The 1-2 Conjecture (multiset version). For every graph $G$,

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

The paper introduces this multiset variant formally and presents it as the easiest of the three total variants; its status is open.