Let $R$ be a domain such that $\bigcap_{n=1}^\infty \mathfrak{a}^n=0$ holds for all proper ideals $\mathfrak{a}$ of $R$ (this holds, for example, if $R$ is Noetherian). Let $K$ be the quotient field of $R$, and let $\overline{R}$ be the integral closure of $R$ in $K$.
Does the relation $\bigcap_{n=1}^\infty \mathfrak{A}^n=0$ hold for all proper ideals $\mathfrak{A}$ of $\overline{R}$?
