Suppose $X$ is an orientable surface with non-empty boundary and $f:X\to X$ is a pseudo-Anosov automorphism that acts identically on $H_1(X,\mathbf{Z})$. Let $x$ be a fixed point of $f$.

For any $\gamma\in\pi_1(X,x)$ we have $\gamma^{-1}f(\gamma)\in [\pi_1(X,x),\pi_1(X,x)]$, the commutant of $\pi_1(X,x)$. More generally, we have $\gamma\cdot g^{-1}f g(\gamma)\in [\pi_1(X,x),\pi_1(X,x)]$ where $g$ is an automorphism of $X$ that fixes $x$.

I would like to ask what one can say about the normal closure in $\pi_1(X,x)$ of the set of all elements $\gamma\cdot g^{-1}f g(\gamma)$ where $\gamma$ runs through $\pi_1(X,x)$ and $g$ runs through the set of all diffeomorphisms $X\to X$ that fix $x$. In particular, does this closure coincide with the commutant of $\pi_1(X,x)$?