[53, Lemma 4.1, 4.2] Given any $\epsilon\ll 1$, and let $t$ be small enough depending on $\epsilon$. There is a Kähler metric $\omega_{\psi,t}$, such that

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  On $\text{Log}_{\mathcal{X}}^{-1}(\text{Int}(\Delta_{J}))$, the local Kähler potentials $\phi_{J,t}$ of $\omega_{\psi,t}$ can be chosen to satisfy $|\phi_{J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}}|<\epsilon$.

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  The total variation $\int_{X_{t}}|\frac{|\log|t||^{n}\omega_{\psi,t}^{n}}{(L^{n})}-d\mu_{t}|<\epsilon.$

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  The Kähler potential of $\omega_{\psi,t}$ relative to a fixed Fubini-Study reference metric, is uniformly bounded independent of $t,\epsilon$.