The residual-intersection conjecture for powers of ideals

Let $S$ be a standard graded polynomial ring over an infinite field, let $I\subset S$ be a homogeneous ideal generated in a single degree $\delta$ with analytic spread $\ell$, and let $J\subset I$ be generated by $\ell-1$ general elements of degree $\delta$. Set $R=S/(J:I)$. The ideal $J:I$ is a geometric $(\ell-1)$-residual intersection of $I$ when it is a residual intersection with the stated geometricity condition, and $I$ is unmixed when all its associated primes have the same dimension.

Residual-intersection conjecture. If $I$ is unmixed and $J:I$ is a geometric $(\ell-1)$-residual intersection of $I$, then, for all $\rho\gg0$,

$$I^{\rho}R$$

is a maximal Cohen–Macaulay $R$-module.

This extends the preceding theorem on asymptotic powers after residual intersection, which establishes the same conclusion under stronger hypotheses, including eventual linear presentation. The conjecture asks whether unmixedness and geometric residual-intersection hypotheses alone guarantee maximal Cohen–Macaulayness for all sufficiently high powers.