In principle, it is possible to obtain more refined estimates with respect to the higher order weighted norms of $\zeta_{\pm}$ and $\phi$ by more direct calculation. The above argument using weighted Schauder estimates avoids the lengthy computations, and it suffices for our purpose since in our setting the error caused by complex structure perturbation is at the scale $e^{-CT^{2}}$ while the weighted analysis in the region $\{|z|\leq 1\}$ only introduces at most $e^{CT}$ error. It is also possible to improve the estimates by working on a scale much smaller than the regularity scale, but again that is not needed for our applications in this paper.