An extremal composition operator on the Hardy space of the bidisk with small approximation numbers

We construct an analytic self-map $\mathrm{\Phi }$ of the bidisk ${\mathbb{D}}^2$ whose image touches the distinguished boundary, but whose approximation numbers of the associated composition operator on $H^2({\mathbb{D}}^2)$ are small in the sense that ${\mathrm{lim\hspace{0.17em}sup}}_{n\to \mathrm{\infty }}[a_{n^2}(C_{\mathrm{\Phi }}){]}^{1\mathrm{/}n}<1$.