Choose a smooth nonnegative bump function $\eta$ on $\mathbb{R}^{n}$ supported in $B(0,2)$, and equals one on $B(0,1)$, and construct $\eta_{i}=\eta(\frac{|x-p_{i}|}{r(\delta)})$ supported on $B_{i}$. We calculate $\text{Log}_{\mathcal{X}}^{-1}(B_{l})$

$$\begin{split}dd^{c}\phi_{0}\circ\text{Log}_{\mathcal{X}}=\frac{1}{4\pi|\log|t||^{2}}\sum_{1\leq i,j\leq n}\frac{\partial^{2}\phi_{0}}{\partial x_{i}\partial x_{j}}\sqrt{-1}d\log z_{i}\wedge d\log\bar{z}_{j}\\ \gtrsim_{\delta}\frac{1}{|\log|t||^{2}}\sum\sqrt{-1}d\log z_{i}\wedge d\log\bar{z}_{j},\end{split}$$

$$dd^{c}\eta_{l}\circ\text{Log}_{\mathcal{X}}\geq-\frac{C}{|\log|t||^{2}}\sum\sqrt{-1}d\log z_{i}\wedge d\log\bar{z}_{j},$$

so if $0<\epsilon\ll 1$ is sufficiently small dependent on $\delta$, we can ensure $(\phi_{0}+2\epsilon\eta_{l})\circ\text{Log}_{\mathcal{X}}$ is psh on $\text{Log}_{\mathcal{X}}^{-1}(B_{l})$.

Consider the potential function on $\text{Log}_{\mathcal{X}}^{-1}(\text{Int}(\Delta_{J}))$

$$\psi_{t}(z)=\max(0,\max_{\text{Log}_{\mathcal{X}}(z)\in B_{l}}\{(\phi_{0}+2\epsilon\eta_{l})\circ\text{Log}_{\mathcal{X}}-\epsilon-\phi_{J,t}\}),$$

with $\phi_{J,t}$ from Lemma 4.1.

Since $|\phi_{J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}}|<\epsilon$, we see that on $\partial B_{l}$ the maximum is never achieved by $(\phi_{0}+2\epsilon\eta_{l})\circ\text{Log}_{\mathcal{X}}-\epsilon-\phi_{J,t}$, so the fact that this term is only locally defined causes no problem. By construction every term is $dd^{c}\phi_{J,t}$-psh, so the maximum $\psi_{t}$ is also $dd^{c}\phi_{J,t}$-psh. Near the boundary of $\text{Int}(\Delta_{J})$ the maximum is achieved by $\psi_{t}=0$, so $\psi_{t}$ globalizes to define a Lipschitz continuous $\omega_{FS,t}$-psh function on $X_{t}$, with $\left\lVert\psi_{t}\right\rVert_{L^{\infty}}\leq 3\epsilon$.

Morever, on $\text{Log}_{\mathcal{X}}^{-1}(W_{\delta})$ the maximum of $\psi_{t}$ is strictly greater than zero. Perturbing the bump function in the construction if necessary, we may assume the locus on $X_{t}$ where the maximum is achieved by at least two terms is a subset of codimension one, and it is automatically closed. So a.e on $\text{Log}_{\mathcal{X}}^{-1}(W_{\delta})$, the metric $\omega_{\psi,t}=\omega_{FS,t}+dd^{c}\psi_{t}$ is smooth and equals $dd^{c}(\phi_{0}+2\epsilon\eta_{l})\circ\text{Log}_{\mathcal{X}}$ for some $l$. We calculate using the regularity estimates on $dd^{c}\phi_{0}$ that

$$\omega_{\psi,t}^{n}=(dd^{c}(\phi_{0}+2\epsilon\eta_{l})\circ\text{Log}_{\mathcal{X}})^{n}=(1+O(\epsilon))(dd^{c}\phi_{0}\circ\text{Log}_{\mathcal{X}})^{n}.$$

Using the real MA equation (8),

$$(dd^{c}\phi_{0}\circ\text{Log}_{\mathcal{X}})^{n}=n!\det(D^{2}\phi_{0})\prod_{i}\frac{1}{4\pi|\log|t||^{2}}\sqrt{-1}d\log z_{i}\wedge d\log\bar{z}_{i},$$

where $\frac{(L^{n})}{n!}d\mu_{0}=\det(D^{2}\phi_{0})dx_{1}\ldots dx_{n}$ determines the constant $\det(D^{2}\phi_{0})$. Comparing with section 3.1, the normalized CY measure $d\mu_{t}$ satisfies

$$\frac{(L^{n})}{n!}d\mu_{t}=(1+O(\frac{1}{|\log|t||}))\det(D^{2}\phi_{0})\prod_{i}\frac{1}{4\pi|\log|t||}\sqrt{-1}d\log z_{i}\wedge d\log\bar{z}_{i}.$$

For sufficiently small $t$ depending on $\epsilon$ and $\delta$, we combine the above to deduce

$$\omega_{\psi,t}^{n}=(1+O(\epsilon))\frac{(L^{n})}{|\log|t||^{n}}d\mu_{t}.$$

The total complex MA measure is $\int_{X_{t}}\omega_{\psi,t}^{n}=\frac{(L^{n})}{|\log|t||^{n}}$, and the contribution from the smooth region in $\text{Log}_{\mathcal{X}}^{-1}(W_{\delta})$ is greater than $(1-\delta)\frac{(L^{n})}{|\log|t||^{n}}$ for very small $\epsilon$ dependent on $\delta$. Thus the measure contribution from the complement must be less than $\delta\frac{(L^{n})}{|\log|t||^{n}}$, namely $\delta$-percent of the total measure. Thus the total variation of the signed measure $d\mu_{t}-\frac{|\log|t||^{n}}{(L^{n})}\omega_{\psi,t}^{n}$ is smaller than $\delta$ for small enough $\epsilon$. ∎