Ngô's fully faithful image conjecture for very central D-modules

Let $\mathcal M(G)^G$ be the category of adjoint-equivariant $\mathcal D$-modules on $G$, and let $\mathcal M(G)^G_{vc}$ be its full subcategory of very central objects, namely those whose Harish-Chandra image is supported on the diagonal substack. Let $\mathcal M(H)^W_{Nil}$ denote the indicated nil-DAHA subcategory, and let $\mathcal{W}h$ be the Whittaker category.

Ngô's very central image conjecture. The Ngô functor defines a fully faithful braided monoidal functor on abelian categories, whose essential image is $\mathcal M(G)^G_{vc}$. Moreover, the essential image of $\tensor*[^{W}]{{\operatorname{Res}}}{}$ restricted to $\mathcal M(G)^G_{vc}$ is $\mathcal M(H)^W_{Nil}$.

This conjecture describes the essential images of the Ngô and parabolic-restriction functors on very central equivariant $\mathcal D$-modules. The supplied text introduces the relevant subcategory and states the claims, but gives no resolution evidence.