Over each of the finitely many contractible conical subregions, after suitable gauge fixing, we have the deviation estimates

$$\left\lVert g^{(1)}-g_{\text{flat}}\right\rVert_{C^{k,\alpha}_{-1}}\leq C,\quad\left\lVert\omega^{(1)}-\omega_{\text{flat}}\right\rVert_{C^{k,\alpha}_{-1}}\leq C,\quad\left\lVert\Omega-\Omega_{\text{flat}}\right\rVert_{C^{k,\alpha}_{-1}}\leq C.$$

Morever the volume form error function $E^{(1)}$ satisfies $\left\lVert E^{(1)}\right\rVert_{C^{k,\alpha}_{-2}}\leq C$, namely the higher order version of quadratic decay estimate.