Huneke's uniform Artin–Rees conjecture

Let $R$ be an excellent Noetherian ring of finite Krull dimension, and let $N\subseteq M$ be finitely generated $R$-modules. Huneke's uniform Artin–Rees conjecture. There exists an integer $k=k(N,M)$ such that for every ideal $I\subseteq R$ and all $n\geq k$,

$$I^{n}M\cap N\subseteq I^{n-k}N.$$

This asks for an Artin–Rees bound uniform in the ideal $I$, while allowing dependence on the ambient modules; the source states that the conjecture is open in this generality.