There is a real-valued relative Gibbons-Hawking potential $\varphi_{2}$ on the disc $\{|\vec{\mu}|_{a}\leq\frac{1}{3}A^{1/2}\}$, such that its second derivatives are given by

$$\begin{cases}\frac{\partial^{2}\varphi_{2}}{\partial\mu_{i}\partial\mu_{j}}=V_{(1)}^{ij}-\tilde{V}_{(1)}^{ij}-\frac{\partial^{2}\varphi_{1}}{\partial\mu_{i}\partial\mu_{j}},\quad i,j=1,2,\\ \frac{\partial^{2}\varphi_{2}}{\partial\eta\partial\bar{\eta}}=-\frac{1}{4}(W_{(1)}-\tilde{W}_{(1)})-\frac{\partial^{2}\varphi_{1}}{\partial\eta\partial\bar{\eta}},\\ \frac{\partial^{2}\varphi_{2}}{\partial\eta\partial\mu_{i}}=\frac{1}{2}(\beta_{i}-\tilde{\beta}_{i})-\frac{\partial^{2}\varphi_{1}}{\partial\eta\partial\mu_{i}},\quad i=1,2.\end{cases}$$

We can demand the estimates in $\{|\vec{\mu}|_{a}\leq\frac{1}{3}A^{1/2}\}$:

$$|\nabla^{k}_{g_{a}}\varphi_{2}|_{g_{a}}\leq C\nu A^{1/4-k/2},\quad k\geq 0.$$