The centralizer of $x_3x_2x_1x_2x_1x_3^{-1}$ in $F_3$ is $\langle x_3x_2x_1x_2x_1x_3^{-1}\rangle$?

Question

The centralizer of $x_3x_2x_1x_2x_1x_3^{-1}$ in $F_3$ is $\langle x_3x_2x_1x_2x_1x_3^{-1}\rangle$.

The centralizer of an element in $F_3$ is the set of elements of $F_3$ that commute with that element. I also believe that, denoting $z=x_3x_2x_1x_2x_1x_3^{-1}$, $C_F(z)$ is cyclic. I'm not sure where to go from here, though.

Is the given statement true?

Answer 1

Apply some automorphisms to simplify the problem. E.g. first apply conjugation by $x_3$. Next apply an automorphism that sends $x_2x_1$ to say $x_1$.

The question will become: what commutes with $x_1^2$ in $F_3$?