Virtual Betti number conjecture for mapping tori with reducible fibers

Let $X=Y\rtimes S^1$ be a $4$-manifold, where $Y$ is a reducible $3$-manifold. Write $vb_1(X)$ for the supremum of the first Betti numbers of finite covers of $X$.

Virtual Betti number conjecture. One has

$$vb_1(X)=\infty$$

unless $Y=S^2\times S^1$ or $Y=\mathbb RP^3\#\mathbb RP^3$.

This conjecture concerns virtual first Betti numbers of mapping tori whose fibers are reducible. The supplied text gives no resolution or partial result, so its status remains open.