Sequence list 1-2-3 Conjecture for arbitrary edge orderings

Let $G$ be a graph with no component isomorphic to $K_2$. Fix an ordering of $E(G)$. A sequence edge $k$-list-weighting assigns to each edge a weight from its independently assigned list of $k$ real numbers; each vertex receives the sequence of weights on its incident edges in the fixed edge order. Let $\operatorname{ch}_\sigma^e(G)$ be the least $k$ such that every edge-list assignment permits a weighting whose vertex sequences distinguish adjacent vertices. The sequence list 1-2-3 Conjecture. If $G$ is nice, then

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

This is the sequence analogue of the list 1-2-3 Conjecture for every ordering of the edge set. The paper introduces it as an open conjecture after proving only larger general bounds.