Maximality of BaireSHIFT on grids

Determine whether BaireSHIFT(Z^n) contains COMPUTABLE(Z^n) and FIID(Z^n) for every integer n at least 2, and hence whether BaireSHIFT(Z^n) is the maximum class in the inclusion poset considered in the paper.

Background

The paper establishes that BaireSHIFT(Zn) strictly contains FFIID(Zn), but its relationship with COMPUTABLE(Zn) and FIID(Zn) remains unresolved. The authors identify BaireSHIFT as a candidate for the largest class in the known hierarchy.

References

Currently, it appears possible that $BaireSHIFT(Zn)$ is the maximum element of this poset. Is it? To wit, do we have $BaireSHIFT(Zn) \supset COMPUTABLE(Zn)$ and $BaireSHIFT(Zn) \supset FIID(Zn)$?

— Separating complexity classes of LCL problems on grids  (2501.17445 - Berlow et al., 29 Jan 2025) in Section 6, subsection “Open problems”