The normal generation bound for the first $\ell^2$-Betti number

Let $G$ be a torsionfree discrete group, and let $g_1,\dots,g_k$ be elements whose normal closure is $G$. The first $\ell^2$-Betti number of $G$ is denoted by $\beta_1^{(2)}(G)$. Normal generation conjecture. If $G$ is normally generated by $g_1,\dots,g_k$, then

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

This conjecture gives a bound on the first $\ell^2$-Betti number in terms of the number of normal generators; the note proves related bounds under residual finiteness and torsion hypotheses, while the torsionfree case is the stated conjectural problem.