Conjecture (Csóka-Lippner-Pikhurko). If $G$ is a graph with each vertex of degree $\le d$ with at most $d-1$ pendant edges properly coloured, then this pre-colouring can be extended to all edges of $G$, using $d+1$ colours in total.
If proved, this will directly give new bounds on questions of Albert (2010) & Marks (2016) on measurable Vizing's theorem.
The main motivation for us stating this conjecture was that, if the conjecture is true, then Vizing's theorem holds for every graphing. Since the latter result was recently proved by Jan Grebik and me (in https://arxiv.org/abs/1905.01716) via a different route, the conjecture is not so interesting now.