The aspherical-manifold virtual fibring conjecture

Let $M$ be a closed aspherical manifold with RFRS fundamental group $G=\pi_1(M)$, and let $\beta_i^{(2)}(G;\mathbb K)$ denote the $i$th $L^2$-Betti number with coefficients in a field $\mathbb K$. Aspherical-manifold virtual fibring conjecture. The following are equivalent:

1. $\beta_i^{(2)}(G;\mathbb K)=0$ for all $i$ and all fields $\mathbb K$;
2. $M$ is virtually fibred.

This is the manifold analogue of the Poincaré-duality-group prediction: vanishing of all $L^2$-Betti numbers should characterize virtual fibering. The source emphasizes that finite presentability of the kernel is the principal additional difficulty and gives no resolution status.