Finitary factors versus measurable factors on free groups

Determine whether FFIID(𝔽_n) equals FIID(𝔽_n) and whether FIID(𝔽_n) equals MEASURE(𝔽_n) for free groups of rank n at least 2; in particular, prove or refute the conjecture that both equalities fail.

Background

Free groups are among the best-understood nonamenable groups in the paper, but the relationships among finitary factors, measurable factors, and measurable solutions on all free actions remain incomplete. The authors explicitly conjecture that both proposed equalities fail.

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 β€œOpen problems”, Question labeled ques:free