$(R,m)\subseteq (S,n)$ is a local extension of rings and $S$ is a finitely generated $R$-module. If $P$ is a prime ideal of $R$ such that $P\subset m^2$ and $P'$ is a prime ideal in $S$ such that $P'\cap R=P$, is $P' \subset n^2$?
Following the counterexample by Georges, I was wondering if this works under additional assumptions. Such as if $S$ is a domain or if $R,S$ are both complete?