The conjecture on J-Artin monomials for superspace coinvariants

Let $[n]=\{1,\dots,n\}$, let $I_n$ be the relevant coinvariant ideal, and for $J\subseteq[n]$ let $f_J$ be the polynomial and ${\mathrm{st}}(J)$ the associated $J$-staircase. Define the set of $J$-Artin monomials by

$${\mathcal A}_n(J)=\{x_1^{a_1}\cdots x_n^{a_n}:a_i<{\mathrm{st}}(J)_i\}.$$

J-Artin monomial conjecture. For every subset $J\subseteq[n]$, the $J$-Artin monomials descend to a basis of

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

Such a basis would provide an explicit, non-generic basis for the commutative quotients used to construct bases of the superspace coinvariant ring. The source gives no resolution status.