In the region $\{\text{dist}_{g_{a}}(\cdot,\mathfrak{D}_{1})\lesssim A^{1/2}\lesssim|\vec{\mu}|_{a}\}\subset M^{+}_{\nu}$, which is far away from $\mathfrak{D}_{2}$ and $\mathfrak{D}_{3}$,

$$\begin{cases}\left\lVert\tilde{\alpha}_{1}-\frac{1}{2\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}\right\rVert_{C^{k,\alpha}_{0,0}}\leq CA^{-1/4}\nu,\\ \left\lVert\tilde{\alpha}_{2}\right\rVert_{C^{k,\alpha}_{0,0}}\leq CA^{-1/4}\nu,\\ \left\lVert\tilde{\alpha}_{3}\right\rVert_{C^{k,\alpha}_{0,0}}\leq CA^{-1/4}\nu.\end{cases}$$

Likewise with the neighbourhood of $\mathfrak{D}_{2}$ and $\mathfrak{D}_{3}$.