Sagan–Swanson's type B Hilbert-series conjecture for the superspace coinvariant ring

Let $\Omega_n=\mathbb{C}[x_1,\ldots,x_n]\otimes\bigwedge\langle\theta_1,\ldots,\theta_n\rangle$ be the superspace, and let $SR^B_n$ be its type $B$ superspace coinvariant ring. Write $[k]_q=1+q+\cdots+q^{k-1}$, define $[n]!!_q=[n]_q[n-2]_q\cdots$, and let $\operatorname{Stir}^B_q(n,k)$ be the type $B$ $q$-Stirling numbers. The variables $q$ and $z$ record bosonic and fermionic degree, respectively. Sagan–Swanson's type B Hilbert-series conjecture. The bigraded Hilbert series is

$$\operatorname{Hilb}(SR^B_n;q,z)=\sum_{k=0}^n [2k]!!_q\cdot \operatorname{Stir}^B_q(n,k)\cdot z^{n-k}.$$

The conjecture gives the expected type $B$ analogue of the previously conjectured Hilbert-series formula for the superspace coinvariant ring. In the supplied paper, the formula is presented before the authors' proof of the type $B$ result, so its resolution should be checked against the paper's theorem.