Consider the above family (4.3) of complex Monge-Ampère equations, with respect to the degenerating background metrics $\omega_{t}$, $t\in(0,1]$. Fix any $q>n$. If $\varphi_{t}$ is a family of $C^{2}$ solution, normalized by $\sup_{X}\varphi_{t}=0$, and if $\|e^{nF_{t}}\|_{L^{1}(\,{\rm log}\,L)^{q}(\omega_{X}^{n})}$ is uniformly bounded in $t$, then $\|\varphi_{t}\|_{L^{\infty}(X)}$ is uniformly bounded in $t$ as well.