FFIID versus FIID for free groups

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

Background

For the one-dimensional group Z, the measurable, FIID, and FFIID classes coincide. For free groups, however, the paper does not resolve whether every measurable factor-of-i.i.d. LCL solution can be replaced by a finitary one, and the authors conjecture that the equality fails.

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 Section 6, subsection “Open problems”