Shift universality for computable and measurable-factor classes

Determine whether BaireSHIFT(Z^n) contains both COMPUTABLE(Z^n) and FIID(Z^n) for every integer n at least 2.

Background

BaireSHIFT(Zn) is a candidate for the largest class in the inclusion diagram presented in the paper. Although the paper proves that the shift is not universal for the full BAIRE class, it leaves open whether every computable or FIID-solvable problem nevertheless has a Baire-measurable solution on the free shift.

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 7, subsection “Open problems”