Measurable unfriendly coloring

Prove or disprove that every bounded-degree measured graph admits a measurable unfriendly coloring, and determine whether every bounded-degree Borel graph admits a Borel unfriendly coloring.

Background

An unfriendly coloring is a two-coloring in which every vertex has a color different from that of at least half of its neighbors. Partial measurable results require strong bounds on the Radon–Nikodym cocycle.