A note on the deductive strength of the Nielsen-Schreier theorem

We show that the Boolean Prime Ideal Theorem (BPI) does not imply the Nielsen-Schreier Theorem (NS) in ZF, thus strengthening the result of Kleppmann from Nielsen-Schreier and the Axiom of Choice that the (strictly weaker than BPI) Ordering Principle (OP) does not imply NS in ZF. We also show that NS is false in Mostowski's Linearly Ordered Model of ZFA+BPI. The above two results also settle the corresponding open problems from Howard and Rubin's Consequences of the Axiom of Choice.