Virtual positive first Betti number conjecture for 3-manifolds

Let $M$ be a closed, irreducible $3$-manifold with infinite $cpi_1$.

Virtual positive first Betti number conjecture. $M$ has a finite cover $ctilde{M}$ with

$$H_1(\tilde{M}, \mathbb{Z})$$

infinite.

The source describes this as a stronger conjecture than the Waldhausen–Thurston virtual Haken conjecture. It concerns whether every such $3$-manifold has a finite cover with infinite first homology; no resolution evidence is supplied in the source.