The ring-structure conjecture for primitive local cohomology

Let $M_{n,k}^{\mathrm{prim}}$ be the primitive quotient and let the preceding primitive vector-space conjecture identify it with $H^*(X)[\sum_{i=1}^nx_i,\sum_{i=1}^ny_i]$. Ring-structure conjecture. This isomorphism is an isomorphism of rings. The claim strengthens the proposed bigraded vector-space identification by requiring compatibility with multiplication; the supplied status evidence resolves a different shuffle conjecture, not this ring-structure claim, so no resolution of the present claim is given.