FIID versus MEASURE for higher-dimensional grids

Determine whether the classes FIID(Z^n) and MEASURE(Z^n) are equal for every integer n at least 2.

Background

The class MEASURE(Zn) consists of LCL problems solvable measurably on every free Borel action, whereas FIID(Zn) concerns measurable solutions on the free part of the Bernoulli shift. The paper establishes separations involving FFIID(Zn), but leaves the relationship between FIID(Zn) and MEASURE(Zn) unresolved.

References

Similarly, it is not known whether $FIID(Zn) = MEASURE(Zn)$ for $n 2$.

— Separating complexity classes of LCL problems on grids  (2501.17445 - Berlow et al., 29 Jan 2025) in Section 1, subsection “Finitary factors of i.i.d.”