We will ignore all exponentially small errors coming from identifying tubular neighbourhoods with normal bundles. We compute $dd^{c}\phi_{D_{1}}$ by the Leibniz rule:

and similarly with $dd^{c}(\tilde{x}_{1}^{\frac{n+2}{n}-\frac{kn}{n-1}}x_{2}^{\frac{kn}{n-1}})$.

Comparing $v_{0}dd^{c}\tilde{x}_{1}^{\frac{n+2}{n}}$ and the term $\frac{2(n+2)v_{0}}{n^{2}d_{2}^{2}}x_{1}^{\prime-\frac{n-2}{n}}\frac{\sqrt{-1}}{4\pi}d\log\xi^{\prime}\wedge d\overline{\log\xi^{\prime}}$, the deviation is of order $O(\frac{|x_{2}|+|x_{1}-x_{1}^{\prime}|}{x_{1}^{\prime}})$. Notice for $n\geq 3$, we have $\frac{n}{2(n-1)}<1$ and so $\frac{|x_{2}|}{x_{1}^{\prime}}\leq\left(\frac{x_{2}}{x_{1}}\right)^{\frac{n}{2(n-1)}}$. For $|x_{1}-x_{1}^{\prime}|\leq x_{1}^{\prime\frac{n-2}{2(n-1)}}$, the error $O(\frac{|x_{1}-x_{1}^{\prime}|}{x_{1}^{\prime}})=O(x_{1}^{-\frac{n}{2(n-1)}})$, which is absorbed into $O((\frac{x_{2}}{x_{1}})^{\frac{n}{2(n-1)}})$.

Using the lemmas, we can estimate the terms appearing in $dd^{c}(\tilde{x}_{1}^{\frac{n-2}{n(n-1)}}\phi_{TY})$ in the local $C^{k,\alpha}$-norms:

The cross terms have order

So does its complex conjugate. The last error comes from the deviation between $\tilde{x}_{1}^{\frac{n-2}{n(n-1)}}dd^{c}\phi_{TY}$ from $x_{1}^{\prime\frac{n-2}{n(n-1)}}dd^{c}\phi_{TY}$. This error is again of order $O(\frac{|x_{2}|+|x_{1}-x_{1}^{\prime}|}{x_{1}^{\prime}})$.

The remainder terms have leading order contribution $dd^{c}(\tilde{x}_{1}^{\frac{n+2}{n}-\frac{2n}{n-1}}x_{2}^{\frac{2n}{n-1}}).$ Its main contribution is $\tilde{x}_{1}^{\frac{n+2}{n}-\frac{2n}{n-1}}dd^{c}x_{2}^{\frac{2n}{n-1}}$, which has magnitude $O((\frac{x_{2}}{x_{1}})^{\frac{n}{n-1}})$. Combining all the errors, the deviation between $dd^{c}\phi_{D_{1}}$ and the local product metric (32) has magnitude bounded by $O((\frac{x_{2}}{x_{1}})^{\frac{n}{2(n-1)}}).$ ∎