(Sketch)

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  The first ingredient is that by the regularity theory of real MA equation (cf. section 4.5), after deleting a subset of $\text{Int}(\Delta_{J})$ of Hausdorff $(n-1)$-measure zero, then $\phi_{0}$ is smooth. After a slight shrinking of the remaining open set, then $\phi_{0}$ has $C^{k}$ bounds.

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  The second ingredient is Savin’s small perturbation theorem (cf. section 4.8). After passing to the local universal cover, both $\phi_{CY,J,t}$ and $\phi_{0}\circ\text{Log}_{\mathcal{X}}$ solve a complex Monge-Ampère equation. The difference in their RHS vanishes in the $t\to 0$ limit in arbitrarily high $C^{k}$ norm, as a consequence of the volume form asymptote in section 3.1. Savin’s result then improves the $C^{0}$ closeness of $\phi_{CY,J,t}$ and $\phi_{0}\circ\text{Log}_{\mathcal{X}}$ to $C^{\infty}_{loc}$ closeness, after small shrinking of $U_{J,t}$.

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