Equality of FFIID and FIID for free groups

Determine whether FFIID(F_n) equals FIID(F_n) for every free group F_n with n at least 2.

Background

The authors explicitly include the comparison between finitary and measurable factor-of-i.i.d. classes for free groups among the unresolved problems, conjecturing that the classes differ.

References

Do we have $FFIID(\mathbb{F}_n) = FIID(\mathbb{F}_n)$ or $FIID(\mathbb{F}_n) = MEASURE(\mathbb{F}_n)$? We conjecture the answer is ``no'' to both.

— Separating complexity classes of LCL problems on grids  (2501.17445 - Berlow et al., 29 Jan 2025) in Question 6.8, Section 6, subsection “Open problems”