Applying the asymptotes in Proposition 4.16 and 4.18, on the local chart $\{r\lesssim A^{1/4}\}$, up to an error of order $O(A^{-3/4}\max(1,\log(A^{-1/4}\varrho)))$, the ansatz metric admits asymptote

(4.21)

$$\begin{cases}\begin{split}g^{(1)}=&V_{(1)}d\mu^{2}+V_{(1)}^{-1}\vartheta^{2}+\text{Re}(W^{p\bar{q}}_{(1)}d\eta_{p}\otimes d\bar{\eta}_{q})\\ \sim&\frac{1}{2\sqrt{\mu^{2}+|\xi_{1}|^{2}}}(d\mu^{2}+|d\xi_{1}|^{2})+(A+\frac{1}{2\sqrt{\mu^{2}+|\xi|^{2}}})^{-1}\vartheta^{2}+g_{a},\end{split}\\ \omega^{(1)}\sim d\mu\wedge\vartheta+\frac{\sqrt{-1}}{2}(a_{p\bar{q}}d\eta_{p}\wedge d\bar{\eta}_{q}+\frac{1}{2\sqrt{\mu^{2}+|\xi_{1}|^{2}}}d\xi_{1}\wedge d\bar{\xi}_{1}),\\ \Omega^{(1)}\sim A^{1/2}\{\vartheta-\sqrt{-1}(A+\frac{1}{2\sqrt{\mu^{2}+|\xi_{1}|^{2}}})d\mu)\}\wedge d\eta_{1}\wedge d\eta_{2}.\end{cases}$$

The main issue then is to compare the connection $\vartheta$ with $\vartheta_{\text{NUT}}$. The curvature of the Taub-NUT metric $d\vartheta_{\text{NUT}}$ has the explicit formula

$$\sqrt{-1}\{\frac{-\mu}{4(\mu^{2}+|\xi_{1}|^{2})^{3/2}}d\xi_{1}\wedge d\bar{\xi}_{1}+\frac{-\bar{\xi}_{1}}{4(\mu^{2}+|\xi_{1}|^{2})^{3/2}}d\mu\wedge d\xi_{1}-\frac{-\xi_{1}}{4(\mu^{2}+|\xi_{1}|^{2})^{3/2}}d\mu\wedge d\bar{\xi}_{1}\}.$$

The curvature $d\vartheta$ is prescribed by formula (1.6), involving the first derivatives of $v$ and $w^{p\bar{q}}$. Applying the asymptotic from Proposition 4.18,

$$|(d\vartheta-d\vartheta_{\text{NUT}})|_{g_{a}}\leq CA^{-1/2},\quad|\nabla_{g_{a}}(d\vartheta-d\vartheta_{\text{NUT}})|_{g_{\text{NUT}}}\leq CA^{-1/4}\ell^{-2}.$$

In particular $\left\lVert(d\vartheta-d\vartheta_{\text{NUT}})\right\rVert_{C^{1}_{-1}(g_{\text{NUT}})}\leq CA^{1/4}$. After suitable gauge fixing, we can find a 1-form $\vartheta-\vartheta_{\text{NUT}}$ on the base $\{r\lesssim A^{1/4}\}$, with norm estimate upstairs $\left\lVert\vartheta-\vartheta_{\text{NUT}}\right\rVert_{C^{1,\alpha}_{0}(g_{\text{NUT}})}\leq CA^{-1/4}$ for any fixed $0<\alpha<1$. This specifies a gauge choice of $\vartheta$.

Thus up to an admissible amount of error we can replace $\vartheta$ by $\vartheta_{\text{NUT}}$ in the asymptote (4.21). The deviation between RHS of (4.21) and $g_{\text{NUT}}$ is an elementary term $\Psi^{*}g_{a}-A(d\mu^{2}+|d\xi_{1}|^{2}+|d\xi_{2}|^{2})$ controlled by (4.19). ∎