The multigraph formulation of the 1-2-3 Conjecture

Let $G$ be a graph, and let $\mathcal{M}^{[3]}(G)$ be the family of multigraphs obtained from $G$ by edge multiplication with edge multiplicities at most $3$. A multigraph is locally irregular when adjacent vertices have distinct multigraph degrees. 1-2-3 Conjecture. For every graph $G$ containing no isolated edges, there exists a locally irregular multigraph $\hat G\in\mathcal{M}^{[3]}(G)$. This is the multigraph formulation of the neighbor-sum-distinguishing 1-2-3 Conjecture, which the source describes as open and equivalent to the preceding formulation.