The unbalanced bipartite measurable edge-coloring conjecture

Let $G=(V,E,\pi)$ be a Borel multigraph of maximum degree $\Delta\in{\mathbb N}$ with an unbalanced bipartition $(A,B)$, meaning that the two sides have unequal relative sizes in the sense used for the paper's measurable matching theorem. For a probability measure $\mu$ on $V$, let $\chi'_\mu(G)$ denote the measurable edge-chromatic number.

Unbalanced bipartite measurable edge-coloring conjecture. For every probability measure $\mu$ on $V$,

$$\chi'_\mu(G)\leqslant\Delta.$$

This conjecture extends the paper's measurable matching theorem. König's theorem gives the same bound combinatorially for bipartite multigraphs, so the issue is whether the edge-coloring can be chosen measurably.