The fibered-cover characterization of locally free subgroups

Let $N$ be a finite-volume hyperbolic $3$-manifold. A locally free subgroup is a subgroup all of whose finitely generated subgroups are free, and a finite cover of $N$ fibers over the circle if it is the total space of a fibration over $S^1$.

Fibered-cover characterization conjecture. The fundamental group $\pi_1(N)$ contains a subgroup which is locally free but not free if and only if $N$ does not have a finite cover which fibers over the circle.

This conjecture aims to use the existence of locally free but nonfree subgroups to distinguish finite-volume hyperbolic $3$-manifolds according to whether they virtually fiber over the circle. The source points to the existence of infinitely many commensurability classes of hyperbolic $3$-manifolds that fiber over the circle, but gives no resolution of the proposed equivalence.