Computing the core of ideals in arbitrary characteristic

Let $R$ be a local Gorenstein ring with infinite residue field of arbitrary characteristic. Let $I$ be an $R$–ideal with $g=\text{height}I>0$, analytic spread $\mathrm{\ell }$, and let $J$ be a minimal reduction of $I$. We further assume that $I$ satisfies $G_{\mathrm{\ell }}$ and ${\depth} R/I^j \geq \dim R/I -j+1$for$1 \leq j \leq \ell-g$. The question we are interested in is whether $\text{core}I=J^{n+1}:\text{ds}{\sum }_{b\in I}(J,b{)}^n$ for $n \gg 0$. In the case of analytic spread one Polini and Ulrich show that this is true with even weaker assumptions (\cite[Theorem 3.4]{PU}). We give a negative answer to this question for higher analytic spreads and suggest a formula for the core of such ideals.