Zabrocki's Hilbert-series conjecture for type A super-diagonal coinvariants

Let $\mathcal{SR}_{\mathfrak{S}_n}^k$ be the $\theta$-degree $k$ component of the super-diagonal coinvariant algebra, and define the $q$-integer $[k]_q=1+q+\cdots+q^{k-1}$ and the $q$-Stirling numbers by

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

Zabrocki's conjecture. For $0\leq k\leq n-1$,

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

This is a specialization of Zabrocki's conjecture for the tri-graded Frobenius series of the type $A$ super-diagonal coinvariant algebra and is consistent with the identity $\operatorname{Hilb}(\mathcal{SR}_{\mathfrak{S}_n};q,-q)=1$.