The conjecture that the marginal ideal sum is prime and perfect

Let $[n]=\{1,\ldots,n\}$ index the variables of a hierarchical model, let $\Delta$ be a simplicial complex on $[n]$, and let $a_1,\ldots,a_n$ be the corresponding state-space sizes. Write $K_{\Delta}$, $Q_{\Delta}$, and $P_{\Delta}$ for the ideals defined in the paper.

Primality and perfection conjecture. If $\Delta$ is any simplicial complex, then

$$K_{\Delta}+Q_{\Delta}=P_{\Delta},$$

where $P_{\Delta}$ is a prime and perfect ideal of grade $1-n+\sum a_i$.

The result is proved when $\Delta$ has three or fewer facets. In addition, the radical of $K_{\Delta}+Q_{\Delta}$ is known to be prime, providing evidence for the conjecture; the general case remains open.