The Hard Lefschetz conjecture for the superspace colon quotients

For $J\subseteq[n]$, let $I_n$ be the relevant coinvariant ideal, let $f_J$ be the associated polynomial, and consider the finite-dimensional graded quotient

$$A_J={\mathbb C}[{\bf x}_n]/(I_n:f_J).$$

A graded algebra has the Hard Lefschetz property if it satisfies Poincare duality and possesses a degree-one element inducing the required multiplication isomorphisms between complementary degrees.

Hard Lefschetz conjecture. For every $J\subseteq[n]$, the quotient ring $A_J$ satisfies the Hard Lefschetz property.

These quotients are complete intersections and therefore satisfy Poincare duality; the conjecture asks for the stronger Lefschetz behavior, with the zero ring included by convention. The source gives no resolution status.