The canonical-module ideal conjecture for polarizations of Artinian monomial ideals

Let $I$ be an Artinian monomial ideal, let $J$ be a polarization of $I$, and let $\Delta(J)$ be the associated simplicial complex. Let $k[\Delta(J)]$ denote its Stanley–Reisner ring and let $\omega_{k[\Delta(J)]}$ denote its canonical module. Canonical-module ideal conjecture. The canonical module $\omega_{k[\Delta(J)]}$ identifies, in a simply described and natural way, as a multigraded ideal of the Stanley–Reisner ring $k[\Delta(J)]$.

Such an identification is known for letterplace ideals, where it gives an explicit description of the boundary of the associated simplicial ball. The general assertion is connected with the characterization of homology balls through their canonical modules, but remains open for arbitrary polarizations.