Minimal model program on the generic fiber of log Calabi-Yau type fibration

We study the minimal model program on the geometric generic fiber of a fibration $f:X\to S$ such that for a Zariski dense subset $S^{\mathrm{\prime }}\subseteq S$, $X_s$ is an $\epsilon$-lc log Calabi–Yau type for every $s\in S^{\mathrm{\prime }}$. We prove that for a fibration $f:X\to S$ of varieties, if the fibers are of $\epsilon$-lc log Calabi–Yau type, then the geometric generic fiber $X_{\bar{\eta }}$ is pklt. In particular, for any big divisor $D$ on $X_{\bar{\eta }}$, we can run the anticanonical MMP and $D$-MMP with scaling of an ample divisor on $X_{\bar{\eta }}$.