The core formula conjecture for ideals satisfying residual conditions

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

Core formula conjecture. The core of $I$ should satisfy

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

This conjecture proposes a uniform description of the core for ideals satisfying the stated residual hypotheses, extending formulas known for more restricted classes such as balanced ideals. Its status is not resolved in the supplied source context.