Conjecture on derived ordinary parts of a Bruhat-cell compact induction

Let $W$ be the Weyl group of $(\mathbf{G},\mathbf{S})$, let $I,J\subseteq\Delta$, let $\sigma$ be a locally admissible smooth representation of $L_I$ over $A$, let ${}^Iw^J\in{}^IW^J$ be a minimal double-coset representative, and let $n\in\mathbb{N}$. Let $d_{{}^Iw^J}$ be the associated integer and $\delta_{{}^Iw^J}$ the associated algebraic character of $\mathbf{L}_{J\cap{}^Iw^J{}^{-1}(I)}$. Derived ordinary-parts conjecture. There is a natural $A[L_J]$-linear isomorphism

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

This would compute derived ordinary parts cell by cell in the Bruhat filtration of parabolic induction, a key step toward understanding derived ordinary parts of induced representations. The statement is presented as a conjecture, and no resolution status is supplied in the source.