Equality of FIID and MEASURE for free groups

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

Background

For free groups, the paper notes known strict separations among several descriptive complexity classes, but the relationship between FIID and MEASURE remains unresolved. The authors state that they conjecture a negative answer.

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”