Equivalence of vNE index 1 and common fundamental domain

Determine whether, for countable discrete groups Γ and Λ, von Neumann equivalence with coupling index 1 is equivalent to von Neumann equivalence with a common fundamental domain; specifically, establish that the existence of a semi-finite von Neumann algebra (M, Tr) with commuting trace-preserving actions of Γ and Λ admitting finite-trace fundamental domains p and q such that Tr(p)/Tr(q) = 1 is equivalent to the existence of a von Neumann coupling (M, Tr) that admits a projection which is simultaneously a fundamental domain for both actions.

Background

The paper introduces von Neumann orbit equivalence (vNOE) for groups as the existence of a von Neumann coupling with a common fundamental domain, paralleling the index-1 condition in measure equivalence for orbit equivalence. For von Neumann equivalence (vNE) couplings between groups Γ and Λ, the index [Γ:Λ]_M is defined as Tr(p)/Tr(q) using chosen fundamental domains p and q. The authors suspect that setting the index to 1 should capture the same phenomenon as having a common fundamental domain, but they do not establish this equivalence.

Resolving this question would clarify whether the newly defined vNOE of groups coincides with the subclass of vNE couplings of index 1, thereby linking two perspectives (index and common domain) on the same equivalence concept.