Bartnicki–Grytczuk–Niwczyk list 1-2-3 Conjecture

Let $G$ be a graph with no component isomorphic to $K_2$. An edge $k$-list-weighting assigns to each edge a weight from an independently assigned list of $k$ real numbers. Let $\operatorname{ch}_\Sigma^e(G)$ be the least $k$ such that every assignment of lists of size $k$ to the edges permits an edge list-weighting giving different incident-edge sums to adjacent vertices. Bartnicki–Grytczuk–Niwczyk's list 1-2-3 Conjecture. If $G$ is nice, then

$$\operatorname{ch}_\Sigma^e(G)\leq 3.$$

This strengthens the ordinary 1-2-3 Conjecture by allowing arbitrary edge lists. The paper states it as a conjecture and develops sequence analogues, so it remains open in the source.