The superspace coinvariant problem

Let $\Omega_n$ be the superspace algebra with commuting variables $x_1,\ldots,x_n$ and anticommuting variables $\theta_1,\ldots,\theta_n$, acted on by the symmetric group $\mathfrak{S}_n$. Let $\langle (\Omega_n)^{\mathfrak{S}_n}_+\rangle\subseteq\Omega_n$ be the ideal generated by the $\mathfrak{S}_n$-invariants with vanishing constant term, and let $\mathbb{W}_{n,k}$ denote the corresponding superspace module. Write $\operatorname{grFrob}(-;q,z)$ for the graded Frobenius characteristic, where $q$ records commuting-variable degree and $z$ records anticommuting-variable degree, and let $\{z^{n-k}\}$ extract the coefficient of $z^{n-k}$. The superspace coinvariant problem. For any $k$, one has

$$\{z^{n-k}\}\,\operatorname{grFrob}\left(\Omega_n/\langle(\Omega_n)^{\mathfrak{S}_n}_+\rangle;q,z\right)=\{z^{n-k}\}\,\operatorname{grFrob}(\mathbb{W}_{n,k};q,z).$$

This is presented as an equivalent formulation of the superspace coinvariant problem that motivates the work. The supplied passage does not state whether the problem has been solved, so its resolution remains open here.