The normal-closure inequality for torsion-free groups and p-groups

Let $G$ be a torsion free discrete group, or a $p$-group for a prime $p$, and let $g_1,\dots,g_n$ be arbitrary elements of $G$. Denote by $\left\langle\!\left\langle g_1,\dots,g_n\right\rangle\!\right\rangle$ the normal closure of these elements in $G$. Normal-closure inequality.

$$\beta_1^{(2)}\left(G/\left\langle\!\left\langle g_1,\dots,g_n\right\rangle\!\right\rangle\right) \geq \beta_1^{(2)}(G)-n.$$

The conjecture concerns how much the first $\ell^2$-Betti number can decrease after imposing $n$ normal relations. The paper explains that coprime torsion is essential to its constructions and notes applications to the Wiegold and Kervaire problems; the conjecture itself remains open.