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.

(This problem was written 23.08.2018 by Oleg Pikhurko on page 51 of Volume 2 of the Lviv Scottish Book).