Finitary factors versus Borel and Baire complexity

Determine the relationship between FFIID(Z^n) and BOREL(Z^n), and determine the relationship between FFIID(Z^n) and BAIRE(Z^n), including whether either class is contained in the other.

Background

The paper proves an inclusion from FFIID(Γ) into a shift-based Baire class for arbitrary finitely generated groups, but it does not settle the corresponding relationships with BOREL(Zn) or BAIRE(Zn). In particular, neither direction of inclusion between FFIID(Zn) and BAIRE(Zn) has been excluded.

References

What is the relationship between $FFIID(Zn)$ and $BOREL(Zn)$? %Is either of these classes a subset of the other one? What about $FFIID(Zn)$ and $BAIRE(Zn)$? No inclusion between these classes has been ruled out yet.

— Separating complexity classes of LCL problems on grids  (2501.17445 - Berlow et al., 29 Jan 2025) in Section 7, subsection “Open problems”