List distinguishing index conjecture for connected graphs

Let $G$ be a connected graph, finite or infinite. For each edge $e\in E(G)$, let $L(e)$ be a list of available colours. The list distinguishing index $D'_l(G)$ is the minimum cardinal number $k$ such that every assignment of lists of cardinality $k$ admits a distinguishing edge colouring $c$ with $c(e)\in L(e)$ for every edge $e\in E(G)$. The distinguishing index $D'(G)$ is the minimum number of colours in a distinguishing edge colouring of $G$.

List distinguishing index conjecture.

$$D'_l(G)=D'(G).$$

This conjecture extends the finite-graph vertex list-distinguishing conjecture to edge colourings and to connected infinite graphs. The source presents it as being motivated by the Ferrara–Flesch–Gethner conjecture and the List Colouring Conjecture; no resolution is given.