Let $G = A\ast B$ be a free product of groups $A\ne1$ and $B\ne1$ without the elements of order $2$. Suppose that $f\in G$ and $f\notin A^g,B^g$ for any $g\in G$ and let $\left \langle \left \langle f \right \rangle \right \rangle $ be the normal closure of $f$ in $G$.
Is it correct that if $f^n\in \left [ G,\left \langle \left \langle f \right \rangle \right \rangle \right ]$ then $n=0$?