Assume $\left\lVert\phi\right\rVert_{C^{0}}$ has a uniform bound independent of $t$. Then after shrinking $U$ by a small amount independent of $t$, we have

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  The convex function $\bar{\phi}$ has a Lipschitz bound $|\bar{\phi}(x)-\bar{\phi}(x^{\prime})|\leq C|x-x^{\prime}|.$

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  There is an upper bound $\phi-\bar{\phi}\leq\frac{C}{|\log|t||^{1/2}}$.

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  On each logarithmic dyadic scale $U_{a}=\{a_{i}\leq\log|z_{i}|\leq 2a_{i},\forall i\}\subset U$, the $L^{1}$-integral

  $$\int_{U_{a}}|\phi-\bar{\phi}|\prod\sqrt{-1}d\log z_{i}\wedge d\overline{\log z_{i}}\leq\frac{C}{|\log|t||^{1/2}}.$$

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  There is an improved Skoda inequality with uniform constants $\alpha,C$ independent of $t$:

  $$\int_{U}e^{-\alpha|\log|t||^{1/2}(\phi-\bar{\phi})}d\mu_{t}\leq C.$$