Li–Ni's virtual Betti number conjecture for reducible 3-manifolds

Let $M$ be a closed oriented reducible $3$-manifold and let $f\colon M\to M$ be a homeomorphism. Its mapping torus is $M\rtimes_f S^1$, and the virtual first Betti number is

$$vb_1(X)=\sup\{b_1(\overline{X})\mid \overline{X}\text{ is a finite cover of }X\}.$$

Li–Ni's conjecture. If $M$ is a closed oriented reducible $3$-manifold, then

$$vb_1(M\rtimes_f S^1)=\infty,$$

unless $M$ is finitely covered by $S^2\times S^1$. The conjecture concerns when mapping tori of reducible 3-manifolds have virtually infinite first Betti number. The source states that this conjecture has been confirmed by Ni using a group-theoretic result.