Equality of Borel and Baire complexity classes on grids

Determine whether the classes BOREL(Z^n) and BAIRE(Z^n) are equal for every integer n at least 2.

Background

The paper proves that BAIRE(Zn) and MEASURE(Zn) are distinct for n at least 2, while the general inclusions place BOREL(Zn) between CONTINUOUS(Zn) and MEASURE(Zn) intersected with BAIRE(Zn). The remaining relationship between BOREL(Zn) and BAIRE(Zn) is therefore unresolved.

References

It remains an open problem to determine whether $BOREL(Zn) = BAIRE(Zn)$.

— Separating complexity classes of LCL problems on grids  (2501.17445 - Berlow et al., 29 Jan 2025) in Section 1, immediately after Theorem 1.2