Thurston–Waldhausen virtually positive Betti number conjecture

Let $M$ be a hyperbolic $3$-manifold. Property $(FAb)$ means that every finite-index subgroup has finite abelianization. Thurston–Waldhausen's VPBN conjecture. The manifold $M$ has a finite cover with positive first Betti number. Equivalently, $\pi_1(M)$ does not have property $(FAb)$; that is, $\pi_1(M)$ has a finite-index subgroup with infinite abelianization. This is presented as one of the main open problems about $3$-manifolds. The source explains that Golod–Shafarevich theory alone cannot settle it because torsion Golod–Shafarevich groups exist.