The torsion-free cyclic relation-module conjecture

Let $F$ be a free group, let $N\triangleleft F$ be a normal subgroup, and set $G=F/N$. Define the rational relation module by

$$N_{\mathbb{Q}}=\mathbb{Q}\otimes_{\mathbb{Z}}N/[N,N].$$

The torsion-free cyclic relation-module conjecture. If $G$ is torsion-free and $N_{\mathbb{Q}}$ is cyclic as a $\mathbb{Q}G$-module, then $N$ is normally generated by a single element.

This conjecture asks whether the relation-gap phenomenon can occur for torsion-free groups with cyclic rational relation module. The preceding example shows only that the analogous assertion fails in the presence of torsion, while the torsion-free case is left open in the source.