Integral closure and local cohomology

Let $A$ be a Noetherian ring and let $I$ be an ideal in $A$. Let $\mathcal{F}=\{J_n{\}}_{n\ge 0}$ be a multiplicative filtration of ideals in $A$ such that $\mathcal{R}(\mathcal{F})={⨁}_{n\ge 0}{J}_n$ is a finitely generated $A$-algebra. Let $\mathcal{R}=A[It]$ and assume $I^n\subseteq J_n$ for all $n\ge 1$. We show the following two assertions are equivalent: (1) For all $i\ge 0$ we have $H_{{\mathcal{R}}_{+}}^i(\mathcal{R}(\mathcal{F}){)}_n=0$ for all $n\gg 0$. (2) $J_n\subseteq \overline{I^n}$ for all $n\ge 1$. Here $\overline{I^n}$ is the integral closure of $I^n$.