Measurable versus factor-of-i.i.d. complexity

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

Background

MEASURE(Zn) is always contained in FIID(Zn), because a measurable solution on every free action yields one on the free part of the shift action. The paper does not determine whether this inclusion is an equality for integer grids of dimension at least two.

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.”; Question 7.3; Section 7, subsection “Open problems”