Relationship between FFIID and BOREL or BAIRE classes

Characterize the inclusion relationship between FFIID(Z^n) and BOREL(Z^n), and between FFIID(Z^n) and BAIRE(Z^n), for every integer n at least 2.

Background

The paper proves that FFIID(Γ) is contained in BaireSHIFT(Γ), but this does not determine its relationship with the broader BOREL and BAIRE classes on grids. The authors state that no inclusion between the relevant classes has yet been ruled out.

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 6, subsection “Open problems”