Stability from weak stability of all quotients

Let $G$ be a group, let $N$ range over the normal subgroups of $G$, and let stability and weak stability refer to the notions for systems of relator words introduced in the paper. Stability conjecture. A group $G$ is stable whenever every quotient $G/N$ is weakly stable. This would clarify the relationship between stability and weak stability in permutations and could provide a way to detect stability through the weak stability of group quotients; the source presents this as an unresolved question.