In the star type region case, by Cor. 4.7, we have the upper bound $\varphi_{m}-\bar{\phi}_{m,w}\leq Cs^{-1/2}$. By Prop. 4.14 and Cor. 4.15, in $U^{s,*}_{w}$ we can replace $\bar{\phi}_{m,w}$ by $\psi_{m}$ up to an error bounded by $Cs^{-1/2}$, hence the claim. The face type region follows the same argument, without the translational invariance statement of Cor. 4.15. ∎