The generators conjecture for primitive local cohomology

Let $M_{n,k}^{\mathrm{prim}}$ be the primitive quotient of the local-cohomology module, and let $\Delta_x=\prod_{i<j\leq n}(x_i-x_j)$. Define

$$\psi_s(\mathrm{pt})=\sum_{i=1}^nx_i^s,\qquad \psi_s(1)=s\sum_{i=1}^nx_i^{s-1}y_i.$$

Generators conjecture. The space $M_{n,k}^{\mathrm{prim}}$ is spanned by products of $1/\Delta_x^{k-1}$ and polynomials of the forms $\psi_s(\mathrm{pt})$ and $\psi_s(1)$. The claim would provide a presentation and ring structure for the primitive local-cohomology space; the supplied text gives no resolution status.