Given any $0<\epsilon\ll 1$, then for sufficiently small $t$ depending on $\epsilon$, there is a smooth Kähler metric $\omega_{FS,t}$ on $(X_{t},\frac{1}{|\log|t||}c_{1}(L))$, such that

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  The relative Kähler potential for any two choices of $\epsilon$ is bounded uniformly independent of small $t$.

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