Let $1\ll\nu\ll A^{3/8}$. Then there is a real valued function $\varphi_{1}$ on $\mathcal{B}^{+}_{\nu}$, solving the generalised Gibbons-Hawking equation on $\mathcal{B}^{+}_{\nu}\cap\{\ell>2A^{1/2}\}$

$$\tilde{V}^{ij}_{(2)}=\tilde{V}^{ij}_{(1)}+\frac{\partial^{2}\varphi_{1}}{\partial\mu_{i}\partial\mu_{j}},\quad\tilde{W}_{(2)}=\tilde{W}_{(1)}-4\frac{\partial^{2}\varphi_{1}}{\partial\eta\partial\bar{\eta}},\quad\det(\tilde{V}^{ij}_{(2)})=\tilde{W}_{(2)}.$$

Morever $\varphi_{1}$ is $\Delta_{a}$-harmonic on $\mathcal{B}^{+}_{\nu}\cap\{\ell<A^{1/2}\}$, and

$$\left\lVert\nabla^{2}_{g_{a}}\varphi_{1}\right\rVert_{C^{k,\alpha}_{-1,0}(\mathcal{B}^{+}_{\nu}\cap\{\ell\gtrsim A^{1/2}\})}\leq C\nu^{3}A^{-3/4},$$

and $|\nabla^{2}_{g_{a}}\varphi_{1}|_{g_{a}}\leq C\nu^{3}A^{-3/2}$ on $\mathcal{B}^{+}_{\nu}$. In particular the matrix $(\tilde{V}^{ij}_{(2)})$ is positive definite and $\tilde{W}_{(2)}$ is positive on $\mathcal{B}^{+}_{\nu}$.