The Poincaré-duality-group virtual fibring conjecture

Let $G$ be a $\mathrm{PD}^n$-group that is RFRS. Here $\beta_i^{(2)}(G;\mathbb K)$ denotes the $i$th $L^2$-Betti number with coefficients in a field $\mathbb K$. Poincaré-duality-group virtual fibring conjecture. The following are equivalent:

1. $\beta_i^{(2)}(G;\mathbb K)=0$ for all $i$ and all fields $\mathbb K$;
2. there exists a finite-index subgroup $G_1\leqslant G$ and an epimorphism $\phi\colon G_1\to\mathbb Z$ such that $\ker\phi$ is a $\mathrm{PD}^{n-1}$-group.

This predicts that, for RFRS Poincaré-duality groups, vanishing of all $L^2$-Betti numbers is exactly the algebraic condition for virtual fibering with a Poincaré-duality-group kernel. The source presents it as an expectation and gives no resolution status.