For $|x_{2}|+1\ll x_{1}$, Lemma 4.11 says that the volume form error of $\phi_{D_{1}}^{(2)}$ is $O(x_{1}^{-\frac{2n}{n-1}}x_{2}^{\frac{1}{n-1}})=O(x_{1}^{-\frac{2n-1}{n-1}})$. In the gluing region $x_{1}\sim\tilde{C}x_{2}$, the volume form error is controlled by the metric gluing error, which is again $O(x_{1}^{-\frac{2n-1}{n-1}})$ by Lemma 4.12. What happens near $D_{2}$ is completely analogous. Finally, in the compact region $x_{1},x_{2}=O(1)$, the smoothness of $\phi_{glue}$ means that the local $C^{k,\alpha}$-norm is $O(1)$. ∎