Local Vizing's theorem

Let $G$ be a graph with a list assignment $L$ to the edges of $G$. For every edge $e=uv$, define its permissible colour list by

$$L(e)=\max\{1,\dots,\max\{\deg(u),\deg(v)\}+1\}.$$

An $L$-edge-colouring assigns to each edge a colour from its list so that incident edges receive distinct colours.

Local Vizing's theorem. There is an $L$-edge-colouring of $G$.

Vizing's theorem gives an edge-colouring from a common list of size $\Delta(G)+1$; this conjecture proposes the corresponding locally varying bound. The source presents it as a belief and an open question, with no resolution supplied.