Complex symmetry of Composition operators induced by involutive Ball automorphisms

Suppose $\mathcal{H}$ is a weighted Hardy space of analytic functions on the unit ball $\mathbb{B}_n\subset\mathbb{C}^n$ such that the composition operator $C_\psi$ defined by $C_{\psi}f=f\circ\psi$ is bounded on $\mathcal{H}$ whenever $\psi$ is a linear fractional self-map of $\mathbb{B}_n$. If $\varphi$ is an involutive Moebius automorphism of $\mathbb{B}_n$, we find a conjugation operator $\mathcal{J}$ on $\mathcal{H}$ such that $C_{\varphi}=\mathcal{J} C^*_{\varphi}\mathcal{J}$. The case $n=1$ answers a question of Garcia and Hammond.