The aperiodic Borel equitable Brooks conjecture

Let $G$ be an aperiodic Borel graph, meaning a Borel graph whose connected components are all infinite, with finite maximum degree $\Delta \geqslant 3$. Let $\mu$ be a $G$-invariant probability measure on $V(G)$. A $\mu$-equitable $\Delta$-coloring is a measurable proper $\Delta$-coloring whose color classes each have $\mu$-measure $1/\Delta$.

Aperiodic Borel equitable Brooks conjecture. The graph $G$ has a $\mu$-equitable $\Delta$-coloring.

The paper proves this under the additional hypothesis $d_\mu(G) \leqslant \Delta/5$, where $d_\mu(G)$ is the average degree, but the unrestricted statement remains open.