The 1-2-3 Conjecture for edge sum-labellings

Let $G$ be a simple graph. For a positive integer $k$, an edge $k$-labelling is a map $\ell:E(G)\to\{1,\dots,k\}$. For each vertex $u$, define its incident sum by

$$\sigma_\ell(u)=\sum_{v\in N(u)}\ell(uv).$$

The labelling is sum-proper when adjacent vertices have distinct incident sums, and $\chi^e_{\rm S}(G)$ is the least such $k$, when it exists. The 1-2-3 Conjecture. If $G$ is connected and $G\neq K_2$, then

$$\chi^e_{\rm S}(G)\leq 3.$$

The conjecture was raised by Karoński, Łuczak, and Thomason in 2004 and was recently solved by Keusch, so its database status is solved.