Weiss hyperfiniteness question for amenable groups

Prove or disprove that the graph generated by every free Borel action of an amenable countable group is hyperfinite, extending the known result for free Borel actions of Z^n.

Background

The paper uses hyperfiniteness for graphs generated by free Borel Zn-actions. It notes that Weiss asked whether an analogous hyperfiniteness theorem holds for every amenable countable group. The question is known to have a positive measure-theoretic analogue when null sets are ignored, but the full Borel version remains unresolved.

References

A long-standing open question raised by Weiss is whether a version of Theorem~\ref{theo:hyperfinite} holds for every amenable countable group in place of $Zn$ ;

— Separating complexity classes of LCL problems on grids  (2501.17445 - Berlow et al., 29 Jan 2025) in Section 2.1, immediately after Theorem 2.4