Bernstein–Zelevinsky duality conjecture for parabolic Orlik–Strauch representations

Let $G$ be the ambient reductive group, let $P$ be a parabolic subgroup of $G$ containing $B$, with Lie algebra $\mathfrak{p}$ and opposite parabolic $\overline{P}$. Let $M$ be a $\mathfrak{g}$-module in the parabolic BGG category $\mathcal{O}^{\mathfrak{p}}$ with algebraic weights, and let $V$ be a finitely presented admissible smooth representation of the Levi factor of $P$. Write $\mathfrak{n}_{\overline{P}}$ for the nilradical of the Lie algebra of $\overline{P}$, and let $\mathbb{D}_{\rm BZ}$ denote Bernstein–Zelevinsky duality. Bernstein–Zelevinsky duality conjecture. One has

$$\mathbb{D}_{\rm BZ}\bigl(\mathcal{F}_{P}^G(M,V)\bigr)=\mathcal{F}_{\overline{P}}^G\bigl(\operatorname{Hom}_E(M,E)^{\mathfrak{n}_{\overline{P}}^{\infty}},\mathbb{D}_{\rm BZ}(V)\bigr).$$

This conjecture extends the calculated duality for locally analytic principal series and for representations constructed from modules in the BGG category $\mathcal{O}$ to parabolic Orlik–Strauch representations. The supplied text gives the preceding theorem-level results motivating the conjecture but does not state a resolution, proof, or disproof of this general assertion.