We choose the charts so that each point on $X_{t}$ is covered by $\leq C$ log scales. Summing over the local Skoda estimates from all log scales, $\int_{X_{t}}e^{-\alpha u}d\mu_{t}$ is bounded by

$$\begin{split}&\frac{C}{|\log|t||^{m}}\sum_{\text{log scales}}\int_{loc}|z_{0}|^{2a_{0}}\ldots|z_{p}|^{2a_{p}}\prod_{1}^{p}\sqrt{-1}d\log z_{i}\wedge d\log\bar{z}_{i}\wedge\prod_{p+1}^{n}\sqrt{-1}dz_{k}\wedge d\bar{z}_{k}\\ &\leq C.\end{split}$$

∎