Dependence of FIID complexity on the base probability space

Determine whether the class FIID(X, μ, Γ) depends on the choice of a nontrivial standard probability space (X, μ) for a finitely generated group Γ, particularly when Γ = Z^n.

Background

The paper defines FIID(X, μ, Γ) as the class of LCL problems admitting a measurable solution on the free part of the shift over (X, μ). For Γ = Z and for nonamenable groups, cited results imply independence from the base space in the relevant settings. The authors ask whether this remains true for groups such as Zn.

References

Assuming $\mu$ is not concentrated on a single point, does the class $FIID(X, \mu, )$ depend on the space $(X,\mu)$? It does not when $ = Z$ ({because $FIID(Z) = MEASURE(Z)$} and when $$ is non-amenable by a result of Bowen Corl.~1.3. But what about, say, $ = Zn$?

— Separating complexity classes of LCL problems on grids  (2501.17445 - Berlow et al., 29 Jan 2025) in Section 6, subsection “Open problems”