Given $0<\epsilon\ll 1$, there is a Kähler metric $\omega^{(3)}=\omega^{(2)}+\sqrt{-1}\partial\bar{\partial}\phi^{(3)}$ with estimate

$$\left\lVert d\phi^{(3)}\right\rVert_{C^{k+1,\alpha}_{-\epsilon,-1+\epsilon}(\mathbb{C}^{3},\Lambda^{1})}\leq CA^{-1/4},$$

such that the volume form error $E^{(3)}$ defined by

$$\frac{3}{4}(E^{(3)}+1)\sqrt{-1}\Omega\wedge\overline{\Omega}=(\omega^{(3)})^{3}$$

satisfies the fast decay estimate $\left\lVert E^{(3)}\right\rVert_{C^{k,\alpha}_{-4-\epsilon,-4+4\epsilon}}\leq C.$ Here the constants only depend on $k,\alpha,\epsilon,\kappa$ and the scale invariant uniform ellipticity bound (2.11). In particular $\omega^{(3)}$ is close to $\omega^{(2)}$ in the $C^{k,\alpha}$-topology outside a compact set, and the volume form error decay rate is faster than quadratic.