We apply Cor. 2.12. The Skoda estimate is verified in Cor. 4.20. The improved Skoda estimate Thm. 4.21 implies an exponential volume decay:

$$\frac{\int_{\varphi-\psi\leq-t}\omega_{\phi}^{n}}{\text{Vol}(X_{s})}\leq Ce^{-\alpha t\sqrt{s}},$$

hence there exists $c\gg 1$, such that for $t_{0}=cs^{-1/2}\log s$,

$$\left(\frac{\int_{\varphi-\psi\leq-t_{0}}\omega_{\phi}^{n}}{\text{Vol}(X_{s})}\right)^{1/2n}\leq Ce^{-\alpha t_{0}\sqrt{s}/2n}=Ce^{-\alpha c\log s/2n}\leq Cs^{-1/2}.$$

Thm 2.7 then implies $\varphi-\psi\geq-Cs^{-1/2}\log s$ as required. ∎