Kielak's algebraic fibring conjecture for virtually residually torsion-free nilpotent groups

Let $G$ be a virtually residually torsion-free nilpotent group. Its first $\ell^2$-Betti number is denoted by $\beta_1^{(2)}(G)$, and $G$ is algebraically fibred if it admits a surjective homomorphism to $\mathbb{Z}$ with finitely generated kernel. Kielak's conjecture.

$$\beta_1^{(2)}(G)=0 \quad\Longleftrightarrow\quad G\text{ is virtually algebraically fibred}.$$

Kielak proved the analogous equivalence for virtually RFRS groups. This conjecture asks whether the RFRS hypothesis can be weakened to virtual residual torsion-free nilpotence; it is motivated by the relationship between vanishing first $\ell^2$-Betti number and algebraic fibring.