The universal locally free subgroup conjecture for finite-volume hyperbolic 3-manifolds

Let $N$ be a finite-volume hyperbolic $3$-manifold. A subgroup is locally free if every finitely generated subgroup of it is free.

Universal locally free subgroup conjecture. The fundamental group $\pi_1(N)$ contains a subgroup which is locally free but not free.

The theorem established in the paper proves this property for infinitely many commensurability classes, while the conjecture proposes it for every finite-volume hyperbolic $3$-manifold. The source notes that there is no direct evidence beyond that theorem and the commensurability invariance of the property.