If a group $G$ has the property that every finitely generated subgroup $H \leq G$ is free, then must $G$ be a free group?

My intuition for thinking that this is true is that a relation on some generators of $G$ always involves only a finite number of generators and therefore would also hold in some subgroup, but I don't see how to make this precise.

locally free.$\endgroup$ – Moishe Kohan Aug 16 at 14:31