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 $\mathcal{B}^{-}_{\nu}\cap\{\tilde{\ell}>2\}$

$$V_{(2)}=V_{(1)}+\frac{\partial^{2}\varphi_{1}}{\partial\mu\partial\mu},\quad W^{p\bar{q}}_{(2)}=W^{p\bar{q}}_{(1)}-4\frac{\partial^{2}\varphi_{1}}{\partial\eta_{p}\partial\bar{\eta}_{q}},\quad\det(W^{p\bar{q}}_{(2)})=V_{(2)}.$$

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

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

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