(Regularisation) Let $\epsilon$ be suffiently small dependent on $\delta$, and construct $\omega_{FS,t}$ as in Lemma 4.1, for $t$ sufficiently small dependent on $\epsilon$ and $\delta$. There is a Lipschitz continuous function $\psi_{t}$ on $X_{t}$ with $\left\lVert\psi_{t}\right\rVert_{L^{\infty}}\leq 3\epsilon$, such that

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  The function $\psi_{t}$ is smooth away from a closed subset with $d\mu_{t}$-measure zero.

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  The (1,1)-current $\omega_{\psi,t}=\omega_{FS,t}+dd^{c}\psi_{t}\geq 0$ is positive on $X_{t}$.

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  The metric estimate $(1-C\epsilon)dd^{c}\phi_{0}\circ\text{Log}_{\mathcal{X}}\leq\omega_{\psi,t}\leq(1+C\epsilon)dd^{c}\phi_{0}\circ\text{Log}_{\mathcal{X}}$ holds on the smooth locus of $\psi$ inside $\text{Log}_{\mathcal{X}}^{-1}(W_{\delta})$.

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