The Stability Conjecture for generic relator tuples in free groups

Fix $k\ge 2$ and $m\ge 1$, and let $F=F(a_1,\dots,a_k)$ be the free group of rank $k$. For an $m$-tuple $\sigma$ of elements of $F$, write $R(\sigma)$ for the corresponding set of relators. The Stability Conjecture. There exists an algorithmically recognizable generic class $\mathcal{Y}$ of $m$-tuples of elements of $F$ such that, whenever $\sigma,\tau\in\mathcal{Y}$ and $\alpha\in\operatorname{Aut}(F)$ satisfy that $R(\sigma)$ and $R(\alpha(\tau))$ have the same normal closure in $F$, one has

$$R(\sigma)=R(\alpha(\tau)).$$

The conjecture would extend the paper's rigidity arguments from generic quotients of the modular group to generic quotients of free groups of arbitrary finite rank. The authors report that their computer experiments support it, but no proof or resolution is given here.