Superspace Vandermonde model for the Delta operator

Let $n=k+r$ and let ${\mathbf a}=(k-1,k-1,\dots,k-1)$ be a length-$r$ sequence whose entries are all $k-1$. Let $\mathcal V_n({\mathbf a})$ be the smallest subspace containing the corresponding superspace Vandermonde and closed under the indicated partial derivatives and polarization operators, and let ${\mathrm {grFrob}}(\mathcal V_n({\mathbf a});q,t)$ be its doubly graded Frobenius image. Superspace Vandermonde conjecture. One has

$${\mathrm {grFrob}}(\mathcal V_n({\mathbf a});q,t)=\Delta'_{e_{k-1}}e_n.$$

The statement proposes a representation-theoretic model for $\Delta'_{e_{k-1}}e_n$; the source reports that its specialization at $t=0$ is proved by Theorem~, while Zabrocki's corresponding conjecture remains open even at $t=0$.