Borel algebraically closed extensions

Determine whether every Borel field has a Borel algebraically closed extension.

Background

The extension must itself be a Borel field and contain the original Borel field through a Borel injective field homomorphism.

References

Does every Borel field have a Borel algebraically closed extension?

— Open Problems in Mathematical Logic  (2608.26628 - Barmpalias et al., 27 Aug 2026) in Section 5