Baire complexity contained in measurable complexity

Determine whether BAIRE(Z^n) is contained in MEASURE(Z^n) for every integer n at least 2, noting that any such inclusion would be strict.

Background

The paper establishes that MEASURE(Zn) is not contained in BAIRE(Zn). Thus, the reverse inclusion is not ruled out, and the authors explicitly ask whether every Baire-measurably solvable LCL problem on Zn is also measurably solvable.

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 7, subsection “Open problems”