Let $F_n$ be the free group generated by $x_i$, for $1\leq i\leq n$. Let $a_i$ be some elements of $F_n$, also for $1\leq i\leq n$. Is there a nice way to tell when the list $\{a_i^{-1}x_ia_i\}$ does not generate $F_n$?

For insufficient reasons partially related to a talk I gave once in Hamburg (video and handout there), and to a paper I'm not done writing (PDF there), I expect that there might be a way to construct out of the $a_i$'s a conjugacy class in $F_n$ (or perhaps in some completion of $F_n$), whose non-triviality implies that $\{a_i^{-1}x_ia_i\}$ do not generate $F_n$. Does this to anyone make sense?

Let $F_n$ be the free group generated by $x_i$, for $1\leq i\leq n$. Let $a_i$ be some elements of $F_n$, also for $1\leq i\leq n$. Is there a nice way to tell when the list $\{a_i^{-1}x_ia_i\}$ does not generate $F_n$?

For insufficient reasons partially related to a talk Balloons and Hoops and their Universal Finite Type Invariant, BF Theory, and an Ultimate Alexander Invariant I gave once in Hamburg (video and handout there), and to a paper I'm not done writing with the same title (PDF there), I expect that there might be a way to construct out of the $a_i$'s a conjugacy class in $F_n$ (or perhaps in some completion of $F_n$), whose non-triviality implies that $\{a_i^{-1}x_ia_i\}$ do not generate $F_n$. Does this to anyone make sense?