Ngô's compatibility conjecture for the quantum Ngô action

Let $G$ be a reductive group with Weyl group $W$, Cartan subgroup $H$, Lie algebra $\mathfrak{h}$, and extended affine Weyl group $W^\textit{aff}\simeq\Lambda_H\rtimes W$. Let $\mathcal{W}h$ denote the category of modules for the spherical nil-DAHA, let $\mathcal D(G)^G$ be the category of adjoint-equivariant $\mathcal D$-modules on $G$, and let $QHR^L$ and $\tensor*[^{W}]{{\operatorname{Ind}}}{}$ be the indicated functors.

Ngô's compatibility conjecture. The following diagram is commutative:

\xymatrix{ QC(\mathfrak{h}^\ast {//} W^\textit{aff}) \ar@{<->}[d]^\wr \ar[r]^{\pi^\ast} & QC(\mathfrak{h}^\ast)^{W^\textit{aff}} \ar@{<->}[d]^\wr \\ Nil_{W^\textit{aff}}{\text{-}}\mathrm{mod} \ar[r] \ar@{<->}[d]^\wr & \ar@{<->}[d]^\wr \mathfrak{D}_H \rtimes W{\text{-}}\mathrm{mod} \ar[d] \ar@/^3pc/[dd]^{\tensor*[^{W}]{{\operatorname{Ind}}}{}} \\ Nil^{sph}_{W^\textit{aff}}{\text{-}}\mathrm{mod} \ar[r] \ar@{<->}[d]^\wr & (\mathfrak{D}_H)^W{\text{-}}\mathrm{mod} \ar[d]_{QHR^L} \\ \mathcal{W}h \ar[r]^{Ng\hat{o}} & \mathcal D(G)^G }

The conjecture asserts compatibility between the Ngô functor, Springer theory, and the Harish-Chandra homomorphism. Its resolution status is not specified in the supplied text.