The multiplicative 1-2-3 Conjecture for multiset-proper edge-labellings

Let $G$ be a graph. A graph is nice if none of its connected components is isomorphic to $K_2$, and let $\chi_{\rm M}(G)$ be the smallest $k\geq 1$ for which $G$ has an m-proper $k$-labelling, meaning an edge-labelling from $\{1,\dots,k\}$ whose incident-label multisets give different values at the ends of every edge.

The multiplicative 1-2-3 Conjecture for multisets. If $G$ is a nice graph, then

$$\chi_{\rm M}(G)\leq 3.$$

This is the multiset counterpart of the original 1-2-3 Conjecture, raised by Addario-Berry, Aldred, Dalal and Reed. Its resolution status is not specified in the supplied material.