Let $F$ be a free group on $x,y,z$. Fix $n>1$ (I am ready to assume that $n$ is large enough). Let $\mathcal{W}$ be the set of *cyclically reduced* words $w$ in $F$ where the letter $z$ appears at least once (i.e., such that $w\notin \langle x,y \rangle$). 


Let $N$ be the normal subgroup of $F$ generated by $\{w^n:w\in\mathcal{W}\}$. 



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> Is the image of $\langle x,y \rangle$ mod $N$ a free group on two generators?
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Remark: if $\mathcal{W}$ is replaced by a single element in it, this follows from the Freiheitssatz.



Free subgroup of a quotient
As observed by Ilya Bogdavov in a comment, the answer is obviously no since the quotient $F/N$ has exponent $n$.


