The module-isomorphism conjecture for generalized coinvariant algebras

Let $n,k,r$ be parameters, let $R_{n,k,r}$ be the graded $S_n$-module quotient defined from $I_{n,k,r}$, and let

$$I'_{n,k,r}:=\langle p_{k+1}({\mathbf{x}}_n),p_{k+2}({\mathbf{x}}_n),\dots,p_{k+n}({\mathbf{x}}_n),e_n({\mathbf{x}}_n),e_{n-1}({\mathbf{x}}_n),\dots,e_{n-r+1}({\mathbf{x}}_n)\rangle$$

be the ideal obtained by replacing the homogeneous symmetric functions in $I_{n,k,r}$ with power sums, and set $R'_{n,k,r}:={\mathbb{Q}}[{\mathbf{x}}_n]/I'_{n,k,r}$. Both quotients carry graded $S_n$-module structures. Module-isomorphism conjecture. There is an isomorphism of graded $S_n$-modules

$$R_{n,k,r}\cong R'_{n,k,r}.$$

This conjecture proposes that replacing the homogeneous symmetric functions by power sums preserves the graded $S_n$-module structure, even though the defining ideals are not generally equal. The quotient with power-sum generators is preferred because it generalizes more readily to two sets of variables and to connections with Macdonald polynomials.