Compactness of composition operators on the Bergman space of the bidisc

Let $\phi$ be a holomorphic self map of the bidisc that is Lipschitz on the closure. We show that the composition operator $C_{\phi }$ is compact on the Bergman space if and only if $\phi (\overline{{\mathbb{D}}^2})\cap {\mathbb{T}}^2=\mathrm{\varnothing }$ and $\phi (\overline{{\mathbb{D}}^2}\setminus {\mathbb{T}}^2)\cap b{\mathbb{D}}^2=\mathrm{\varnothing }$.