The normal-rank bound for torsion-free groups

Let $G$ be a torsion free discrete group. Write ${\rm nrk}(G)$ for its normal rank, the least number of elements needed to normally generate $G$. Normal-rank bound.

$$\beta_1^{(2)}(G) \leq {\rm nrk}(G)-1.$$

The authors present this as a stronger version of the basic inequality for the first $\ell^2$-Betti number. They indicate that coprime torsion is the only known source of counterexamples, leaving the torsion-free case open.