Breuil's locally analytic socle conjecture for definite unitary groups

Let $T_p$ be the diagonal torus of the product of the local groups at $p$, let $E$ be a coefficient field, and retain the notation for the sets $\mathcal S_2^{|S_p|}$, $\Sigma(z_s)_{\widetilde v}$, the companion characters $(\delta_s)_J^c$, and the completed space $\widehat S(U^p,E)_{\overline\rho}[\mathfrak m_\rho]$. For each $s\in\mathcal S_2^{|S_p|}$, let $z_s$ be the corresponding classical point and let $\Sigma(z_s)_{\widetilde v}$ be the set of embeddings for which the associated character is a trianguline parameter.

Breuil's conjecture. For a continuous character $\chi$ of $T_p$ over $E$,

$$I(\chi)\hookrightarrow \widehat S(U^p,E)^{\operatorname{an}}_{\overline\rho}[\mathfrak m_\rho]$$

if and only if there exist $s\in\mathcal S_2^{|S_p|}$ and $J\subseteq\Sigma(z_s)_{\widetilde v}$ such that

$$\chi=(\delta_s)_J^c\delta_{B_p}^{-1}.$$

This is the locally analytic socle formulation of the companion-points conjecture for the completed cohomology attached to the definite unitary group. The supplied source gives no resolution status for this formulation.