The 1-2-3 conjecture for neighbor sum distinguishing edge weightings

Let $G$ be a nice graph, meaning a simple undirected graph with no component isomorphic to $K_2$. For an integer edge weighting $w:E(G)\rightarrow\{1,2,\ldots,k\}$, define

$$\sigma^{e}(v):=\sum_{e\ni v}w(e).$$

The weighting is neighbor sum distinguishing when adjacent vertices receive distinct values of $\sigma^{e}$, and $\chi^{e}(G)$ is the least such $k$. The 1-2-3 conjecture. For every nice graph $G$, $\chi^{e}(G)\leq 3$. The conjecture asserts the best possible universal bound, since some nice graphs require weight 3. Its validity remains open in the source.