1-2-3 Conjecture on neighbor-sum-distinguishing edge-weightings

Let $G$ be a graph with no isolated edge. A neighbor-sum-distinguishing (nsd) $3$-edge-weighting is a mapping from $E(G)$ to $\{1,2,3\}$ such that the sums of the weights incident to any two adjacent vertices are distinct. 1-2-3 Conjecture. Every graph with no isolated edge admits an nsd $3$-edge-weighting. The conjecture is a central graph-labeling problem; the source notes that an nsd $5$-edge-weighting is known, but does not state that the conjecture has been resolved.