Let $F_n$ be a free group on $n$ generators. Let $w \in F_n$ be a word such that there does not exist any solution in $F_n$ for the equation $w.w(t_1, \ldots, t_n) = 1$, where $t_1, \ldots, t_n$ are variables and $w(t_1, \ldots, t_n)$ is the formal image of word on variables $t_1, \dots, t_n$.

For any $v_1, \ldots, v_n \in F_n$ with the property that they generate rank $n$ subgroup, would the equation $w(v_1, \ldots, v_n).(w(v_1, \ldots, v_n))(t_1, \ldots, t_n) = 1$ still have no solution in $F_n$?

I have tried many examples for words in $F_2$ and it works fine. For partial answer in $F_2$ or similarly for $F_n$, one may look this link for similar question.