Finitary and measurable factors of i.i.d. on free groups

Determine whether FFIID(\mathbb{F}_n) equals FIID(\mathbb{F}_n) and whether FIID(\mathbb{F}_n) equals MEASURE(\mathbb{F}_n) for every n at least 2.

Background

For free groups, the paper discusses known separations among several descriptive complexity classes, but the relationships involving finitary factors of i.i.d., measurable factors of i.i.d., and MEASURE(\mathbb{F}_n) remain unresolved. The authors explicitly conjecture negative answers to both equality questions.

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 7.8, Section 7, subsection “Open problems”