Measurable unfriendly coloring for bounded-degree measured graphs

Prove that every measured graph of bounded degree admits a measurable unfriendly 2-coloring, or construct a counterexample; in particular, determine whether every bounded-degree Borel graph admits a Borel unfriendly coloring.

Background

An unfriendly coloring assigns one of two colors so that every vertex has at least half of its neighbors in the opposite color. The paper cites a positive result under strong bounds on the Radon–Nikodym cocycle but states that the unrestricted measurable and Borel cases remain unresolved.

References

Does every measured graph of bounded degree admits a measurable unfriendly coloring? In fact, it is also open whether every Borel graph of bounded degree admits a Borel unfriendly coloring.

— From descriptive to distributed  (2502.15347 - Grebík et al., 21 Feb 2025) in Problem immediately following Problem 6.2, Section 6 (LCL problems)