The radicality conjecture for hierarchical-model ideals

Let $\Delta$ be any simplicial complex, and let $I_{\Delta}$, $Q_{\Delta}$, and $J_{\Delta}$ be the ideals defined in the paper.

Radicality conjecture. The ideal $I_{\Delta}$ is radical if and only if

$$Q_{\Delta}=J_{\Delta}.$$

The conjecture concerns the radicality of ideals associated with simplicial complexes. The supplied context does not state a resolution in general; the preceding results establish radicality in the case where $\Delta$ has fewer than three facets, so the general equivalence remains open.