The comparison principle (see [K] ,[EGZ]) yields for any $s>0$ and $\tau\in[0,1]$

$$\tau^{n}\frac{\mathrm{Cap}_{\omega_{t}}(\{\varphi_{t}\leq-s-\tau\})}{V_{t}}\leq\int_{\{\varphi_{t}\leq-s\}}\frac{(\omega_{t}+dd^{c}\varphi_{t})^{n}}{V_{t}}.$$

It is now an exercise to derive from this inequality an a priori $L^{\infty}-$estimate,

$$\|\varphi_{t}\|_{L^{\infty}(X)}\leq C_{5}+\ {s}_{0}(\omega_{t}),$$

where ${s}_{0}(\omega_{t})$ (see [EGZ],[BGZ]) is the smallest number $s>0$ satisfying the condition $e^{n}C_{0}^{n}\mathrm{Cap}_{\omega_{t}}(\{\psi\leq-s\})/V_{t}\leq 1$ for all $\psi\in PSH(X,\omega_{t})$ such that $\sup_{X}\psi=0$. Recall from ([GZ 1], Prop. 3.6) that

$$\frac{\mathrm{Cap}_{\omega_{t}}(\{\psi\leq-s-\tau\})}{V_{t}}\leq\frac{1}{s}\left(\int_{X}(-\psi)\frac{\omega_{t}^{n}}{V_{t}}+n\right).$$

Since $\frac{\omega_{t}^{n}}{V_{t}}\leq C_{1}\omega_{1}^{n}$, it follows that

$$\frac{\mathrm{Cap}_{\omega_{t}}(\{\psi\leq-s-\tau\})}{V_{t}}\leq\frac{1}{s}\left(C_{1}\int_{X}(-\psi)\omega_{1}^{n}+n\right).$$

Since $\psi$ is $\omega_{1}-$psh and normalized, we know that there is a constant $A=A(X,\omega_{1})>0$ such that $C_{1}\int_{X}(-\psi){\omega_{1}^{n}}\leq A$ for any such $\psi$. Therefore $s_{0}(\omega_{t})\leq s_{0}:=e^{n}C_{0}^{n}(A+n)$ for any $t\in]0,1]$. Finally we obtain the required uniform estimate for all $t\in]0,1]$.