Waldhausen's virtual positive first Betti number conjecture for 3-manifolds

Let $M$) be a compact, connected, $P^2$-irreducible $3$-manifold with infinite fundamental group. A finite cover of $M$ is a covering space $\tilde M$ equipped with a finite-sheeted covering map $\tilde M\to M$, and $b_1(\tilde M)$ denotes its first Betti number.

Waldhausen's conjecture. There is a finite cover $\tilde M\to M$ with $b_1(\tilde M)>0$.

This is presented as a variant of a conjecture of Waldhausen and would provide virtual orderability consequences for irreducible 3-manifold groups. The source gives no resolution status.