The refined type A Hilbert-series conjecture

Let $\mathcal{SR}_{\mathfrak{S}_n}$ be the type $A$ super-diagonal coinvariant algebra, with Hilbert series $\operatorname{Hilb}(\mathcal{SR}_{\mathfrak{S}_n};q,z)$ tracking $x$-degree by $q$ and $\theta$-degree by $z$. Let $\operatorname{Stir}_q(n,k)$ be the $q$-Stirling numbers defined by

$$\operatorname{Stir}_q(n,k)=\operatorname{Stir}_q(n-1,k-1)+[k]_q\operatorname{Stir}_q(n-1,k),\qquad \operatorname{Stir}_q(1,k)=\delta_{k=1}.$$

Refined type A Hilbert-series conjecture. For $1\leq j\leq n-1$,

$$\operatorname{Hilb}(\mathcal{SR}_{\mathfrak{S}_n};q,-q^j)=\sum_{k=0}^n(-q^j)^k[n-k]_q!\operatorname{Stir}_q(n,n-k).$$

This formulation is stated as equivalent to the type $A$ Hilbert-series conjecture and suggests an approach through complexes with the appropriate Euler characteristic.