The nonadjacent maximum-degree measurable edge-coloring conjecture

Let $G=(V,E,\pi)$ be a simple Borel graph of maximum degree $\Delta\in{\mathbb N}$. Assume that no two vertices of degree $\Delta$ in $G$ are adjacent. For a probability measure $\mu$ on $V$, let $\chi'_\mu(G)$ denote the measurable edge-chromatic number.

Nonadjacent maximum-degree measurable edge-coloring conjecture. For every probability measure $\mu$ on $V$,

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

The paper presents this as a stronger conjectural measurable analogue of the classical edge-coloring bound for simple graphs. Its status is open.