The socle-filtration conjecture for ordinary representations

Let $\rho:{\operatorname{Gal}}({\overline{\mathbb Q}}_p/{\mathbb Q}_p)\longrightarrow \widehat{B}_{C_\rho}(E)\subseteq\widehat{G}(E)$ be generic, with $C_\rho\subseteq R^{+\vee}$ closed and minimal under conjugation by $\widehat{B}(E)$, and let $w_{C_\rho}\in W_{C_\rho}$. Let $\chi_\rho$ and $\theta$ be the character and twisting element associated to $\rho$. Socle-filtration conjecture. There exists a unique admissible unitary continuous representation $\Pi(\rho)_{C_\rho,w_{C_\rho}}$ of $G({\mathbb Q}_p)$ over $E$ with socle filtration $0={\rm Fil}_{-1}\Pi(\rho)_{C_\rho,w_{C_\rho}}\subsetneq {\rm Fil}_0\Pi(\rho)_{C_\rho,w_{C_\rho}}\subseteq\cdots$ such that, for $j\in{\mathbb Z}_{\geq0}$,

$${\rm Fil}_{j}\Pi(\rho)_{C_\rho,w_{C_\rho}}/{\rm Fil}_{j-1}\Pi(\rho)_{C_\rho,w_{C_\rho}}\cong\bigoplus_{\substack{I\subseteq w_{C_\rho}(S^\vee)\cap C_\rho\vert I\vert=j}}\Big(\operatorname{Ind}_{B^-({\mathbb Q}_p)}^{G({\mathbb Q}_p)}\Big(\big(\prod_{\alpha\in I^\vee}s_\alpha\big)w_{C_\rho}\Big)^{-1}(\chi_\rho)\cdot(\varepsilon^{-1}\circ\theta)\Big)^{{\mathcal C}^0},$$

where $I$ ranges over subsets of $w_{C_\rho}(S^\vee)\cap C_\rho$ consisting of pairwise orthogonal roots. This conjecture specifies the entire socle filtration and its principal-series graded pieces; the source presents it as a more precise conjectural description following a conjectural description of irreducible constituents. Its resolution is not stated.