In the above situation, given any $\delta>0$ and $t<1$, there exists an open set $U_{t,\delta}\subset X$ such that its diameter with respect to $\omega_{t}$ is less than $C_{1}\delta^{-1/2}$ and its volume with respect to $\omega_{0}$ is at least $\int_{X}\omega_{0}^{n}-\delta$, where $C_{1}$ is a constant independent of $t,\delta$.