On the top-dimensional $L^2$-Betti number of residually poly-$\mathbb{Z}$ groups

Let $G$ be a residually poly-$\mathbb{Z}$ group of finite type. We prove that $G$ admits a poly-$\mathbb{Z}$ quotient with kernel $N$ satisfying ${\mathrm{c}\mathrm{d}}_{\mathbb{Q}}(N)<{\mathbb{c}\mathbb{d}}_{\mathbb{Q}}(G)$ if and only if the top-dimensional $L^2$-Betti number of $G$ vanishes.