Let $-3<\delta<-1$ and $1\ll\nu\ll A^{3/8}$. Let $f$ be a $T^{2}$-invariant function compactly supported in $M^{+}_{\nu}$ with $\left\lVert f\right\rVert_{C^{k,\alpha}_{\delta}}=1$. Then there is a $T^{2}$-invariant function $u$ such that the Poisson equation is approximately solved on $M^{+}_{\nu}$:

$$\left\lVert\Delta_{\tilde{g}^{(4)}}u-f\right\rVert_{C^{k,\alpha}_{\delta}}\ll 1,$$

with the Hessian bound

$$\left\lVert\nabla^{2}_{\tilde{g}^{(2)}}u\right\rVert_{C^{k,\alpha}_{\delta}}\leq C,\quad\left\lVert du\right\rVert_{C^{k+1,\alpha}_{\delta+1}(M^{+}_{\nu})}\leq CA^{-1/4}.$$

The constants depend only on $k,\alpha,\delta,\kappa$ and the scale invariant uniform ellipticity constant of $a_{ij}$.