The core formula conjecture for ideals

Let $R$ be a local Cohen–Macaulay ring with infinite residue field. Let $I$ be an $R$-ideal of analytic spread $ell$ satisfying $G_{\ell}$ and weakly $(\ell-1)$-residually $S_2$. Let $J$ be a minimal reduction of $I$, and let $r$ be the reduction number of $I$ with respect to $J$.

Core formula conjecture. One has

$$\operatorname{core}(I)=(J^r:I^r)I=(J^r:I^r)J=J^{r+1}:I^r.$$

This conjecture gives a formula for computing the core of an ideal in terms of a minimal reduction and its reduction number. The surrounding discussion identifies it as a general conjecture from [CPU2, Conj. 5.1]; the supplied text does not establish whether it has since been proved or disproved.