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Equivalence of Nielsen–Schreier to the axiom of choice

Ascertain whether the Nielsen–Schreier Theorem is equivalent to the axiom of choice in ZF (set theory without choice), i.e., prove or refute that Nielsen–Schreier implies choice or vice versa within ZF.

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Background

The Nielsen–Schreier Theorem states that every subgroup of a free group is free. Known proofs assume some form of choice, and the precise logical strength of the theorem relative to ZF has been a longstanding question in foundations.

References

Even today, all known proofs use the axiom of choice, but it remains an open problem whether or not the Nielsen--Schreier Theorem is equivalent to the axiom of choice (in ZF).

— The theory of one-relator groups: history and recent progress (2501.18306 - Linton et al., 30 Jan 2025) in Subsection 2.3 (The Nielsen–Schreier Theorem, amalgamated free products)