Baire versus MEASURE complexity on grids

Determine whether BAIRE(Z^n) is contained in MEASURE(Z^n) for every integer n at least 2, and characterize whether the inclusion is necessarily strict.

Background

The paper proves that MEASURE(Zn) is not contained in BAIRE(Zn), so any reverse inclusion, if true, would be strict. The authors leave unresolved whether every Baire-measurably solvable LCL problem on Zn is also measurably solvable on all free actions.

References

Does the inclusion $BAIRE(Zn) \,\subset\, MEASURE(Zn)$ hold? ({If it does, it must be strict.}

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