Geometric duality conjecture for Whittaker sheaves

Fix a regular weight $\lambda\in\mathfrak{h}^*$ and let $X$ be the flag variety of $\mathfrak{g}$. Let $Q$ be an open $B$-orbit in a $P_\eta$-orbit on $X$, let $\mathcal{O}_Q$ be an $\eta$-twisted $N$-equivariant connection on $Q$, and let $i:Q\hookrightarrow X$ be the inclusion. Fix a basis $\{\delta_w\}$ of $\mathbb{C}[W_\eta]$ and a $W_\eta$-equivariant isomorphism from $\mathbb{C}[W_\eta]$ to the $W_\eta$-coinvariants of $S(\mathfrak{h})$. If $h_w$ is the image of $\delta_w$ and

$$\beta=\sum_{W_\eta}\lambda(h_w)\delta_w,$$

then geometric duality conjecture. As $U(\mathfrak{g})$-modules,

$$\Gamma(X,i_+\mathcal{O}_Q)^{\vee_\beta}\cong\Gamma(X,i_!\mathcal{O}_Q).$$

More generally, for a holonomic $(\mathcal{D}_\lambda,N,\eta)$-module $\mathcal{M}$ on $X$,

$$\Gamma(X,\mathcal{M})^{\vee_\beta}\cong\Gamma(X,\mathbb{V}_X\mathcal{M}).$$

This proposes a relationship between the algebraic duality defined on Whittaker modules and the geometric distinction between extension by direct image and extension by zero; the general statement remains open in the source.