The q-deformed diagonal harmonics conjecture for G(m,n)

Let $W=G(m,n)$, let $\ell\in\mathbb{N}$, and let $q$ be formal. Define $\mathcal{D}W^{(\ell)}_q$ as the space of polynomials annihilated by all operators $D_{q,\mathbf{d}}$ for which $|\mathbf{d}|$ is divisible by $m$, and let $\mathcal{D}W^{(\ell)}$ be the corresponding undeformed diagonal-harmonic space. The q-deformed diagonal harmonics conjecture. The spaces $\mathcal{D}W^{(\ell)}_q$ and $\mathcal{D}W^{(\ell)}$ are isomorphic as graded $W$-modules. In particular, their Hilbert series agree:

$$\mathcal{D}W^{(\ell)}_q(\mathbf{t})=\mathcal{D}W^{(\ell)}(\mathbf{t}).$$

This conjecture predicts that the formal $q$-deformation preserves the graded $W$-module structure and Hilbert series of the diagonal-harmonic space.