Conjecture on the natural morphism for derived ordinary parts

Let $I,J\subseteq\Delta$, let $\sigma$ be a locally admissible smooth representation of $L_I$ over $A$, let ${}^Iw^J\in{}^IW^J$, and let $n\in\mathbb{N}$. Let $d_{{}^Iw^J}$ and $\delta_{{}^Iw^J}$ be the integer and algebraic character associated with ${}^Iw^J$. Derived ordinary-parts morphism conjecture. The natural $L_J$-equivariant morphism

$$\operatorname{Ind}_{L_J\cap P_{J\cap{}^Iw^J{}^{-1}(I)}^-}^{L_J}\left(\left(\mathrm{H}^{n-[F:\mathbb{Q}_p]d_{{}^Iw^J}}\operatorname{Ord}_{L_I\cap P_{I\cap{}^Iw^J(J)}}\sigma\right)^{{}^Iw^J}\otimes\left(\omega^{-1}\circ\delta_{{}^Iw^J}\right)\right)\longrightarrow\mathrm{H}^{n}\operatorname{Ord}_{P_J}\left(\operatorname{c-ind}_{P_I^-}^{P_I^-{}^Iw^JP_J}\sigma\right)$$

is an isomorphism. This is the strengthened cellwise assertion underlying the computation of derived ordinary parts: it promotes the naturally constructed morphism from an injection on a graded piece to an isomorphism after parabolic induction. The source supplies no evidence resolving the conjecture.