Weiss hyperfiniteness for amenable groups

Prove or refute that the orbit graph of every free Borel action of every amenable countable group on a standard Borel space is hyperfinite.

Background

The paper invokes hyperfiniteness for graphs generated by free Borel Zn-actions and notes that Weiss asked whether an analogous theorem holds for every amenable countable group. The question has positive measure-theoretic analogues but remains open in the Borel setting in the form stated.

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, subsection “Rectangular toasts and measure theory”

Is the orbit equivalence relation of Borel action of countable amenable group hyperfinite?

— Open Problems in Mathematical Logic  (2608.26628 - Barmpalias et al., 27 Aug 2026) in 2025 Section 13