We revisit the calculations in Lemma 4.10. The volume form $(dd^{c}\phi_{D_{1}}^{(2)})^{n}$ should be viewed as a perturbation of $(dd^{c}\phi_{D_{1}})^{n}$. Again $(dd^{c}\phi_{D_{1}}^{(2)})^{n}$ is a convergent power series of $\tilde{x}_{1}^{-\frac{n}{n-1}}$, with coefficient in top degree forms on $D_{1}\setminus D_{2}$. The leading order contribution to $(dd^{c}\phi_{D_{1}}^{(2)})^{n}-(dd^{c}\phi_{D_{1}})^{n}$ is

$$n(n-1)v_{0}dd^{c}\tilde{x}_{1}^{\frac{n+2}{n}}\wedge(a(\frac{d_{1}}{d_{2}})^{\frac{n}{n-1}}\tilde{x}_{1}^{\frac{n-2}{n(n-1)}}dd^{c}\phi_{TY})^{n-2}\wedge\tilde{x}_{1}^{\frac{n+2}{n}-\frac{2n}{n-1}}dd^{c}U,$$

which after some calculation gives

$$\frac{2(n+2)(n-1)}{nd_{2}^{2}}v_{0}a^{n-2}(\frac{d_{1}}{d_{2}})^{\frac{n(n-2)}{n-1}}(dd^{c}U)\wedge\omega_{TY}^{n-2}\wedge\tilde{x}_{1}^{-\frac{n}{n-1}}\frac{\sqrt{-1}}{4\pi}d\log\xi^{\prime}\wedge d\overline{\log\xi^{\prime}}.$$

This is by construction $-\sqrt{-1}d\log\xi^{\prime}\wedge d\overline{\log\xi^{\prime}}\wedge\mathcal{I}_{1},$ which is precisely designed to cancel the leading order error $\sqrt{-1}d\log\xi^{\prime}\wedge d\overline{\log\xi^{\prime}}\wedge\mathcal{I}_{1}$ in (33).

The next order of error has $\tilde{x}_{1}^{-\frac{2n}{n-1}}$ in front. Within the $x_{2}=O(1)$ region, the volume error is now $O(\tilde{x}_{1}^{-\frac{2n}{n-1}})$. For $x_{2}\gg 1$ one needs to be careful about the effect of $U$ having a growing term $a^{\prime}u_{0}$, which will damage the exponential decay. Recall from section 4.3 that $u_{0}=x_{2}^{\frac{1}{n-1}}$ outside some compact region in the Tian-Yau space. The largest new contributions to the volume forms error come from terms such as

$$(\tilde{x}_{1}^{\frac{n-2}{n(n-1)}}dd^{c}\phi_{TY})^{n-2}\wedge d^{c}\tilde{x}_{1}^{\frac{n-2}{n(n-1)}}\wedge d\phi_{TY}\wedge d\tilde{x}_{1}^{\frac{n+2}{n}-\frac{2n}{n-1}}\wedge d^{c}u_{0},$$

whose local $C^{k,\alpha}$-norm is $O(\tilde{x}_{1}^{-\frac{2n}{n-1}}x_{2}^{\frac{1}{n-1}})$. ∎