Breuil–Herzig's ordinary parts conjecture for principal series

Let $F$ be a finite extension of $\mathbb{Q}_p$, let $G$ be a connected reductive group over $F$, and let $P\subset G$ be a standard parabolic subgroup with standard Levi subgroup $L$. Write $B_L=B\cap L$ and $B_L^-=B^-\cap L$, let $W_L$ be the Weyl group of $(L,T)$, and choose Kostant representatives $\widetilde{W}_P\subset W$ for $W/W_L$. For $\widetilde{w}_P\in\widetilde{W}_P$, let $\ell(\widetilde{w}_P)$ be its length and let $\alpha_{\widetilde{w}_P}$ be the sum of the positive roots $\alpha$ such that $\widetilde{w}_P(\alpha)$ is not positive. Let $A$ be a local Artinian $\mathcal{O}_E$-algebra with residue field $k_E$, let $\omega:F^\times\to A^\times$ be the image of the fixed uniformizer character, and let $U$ be a smooth locally admissible representation of $T(F)$ on $A$. Breuil–Herzig's ordinary parts conjecture. For every $n\in\mathbb{N}$, there is a natural $L(F)$-equivariant isomorphism

$$\mathrm{H}^{n}\!\operatorname{Ord}_{P(F)}\left(\operatorname{Ind}^{G(F)}_{B^-(F)}U\right)\cong\bigoplus_{[F:\mathbb{Q}_p]\cdot\ell(\widetilde{w}_P)=n}\operatorname{Ind}^{L(F)}_{B_L^-(F)}\left(U^{\widetilde{w}_P}\otimes(\omega^{-1}\circ\alpha_{\widetilde{w}_P})\right).$$

The conjecture gives the higher ordinary parts of a principal series in terms of principal series for the Levi subgroup. The paper proves the asserted formula in certain cases and uses it to study extensions, while the general statement remains open in the source.