Breuil's companion-points conjecture for definite unitary eigenvarieties

Let $T_p$ be the relevant diagonal torus, let $\mathcal E(U^p)_{\overline\rho}$ be the eigenvariety attached to the definite unitary group, and retain the notation for $\mathcal S_2^{|S_p|}$, the classical points $z_s=(\mathfrak m_\rho,\delta_s)$, the subsets $\Sigma(z_s)_{\widetilde v}$, the companion characters $(\delta_s)_J^c$, and the weight notation $\underline\lambda_{\Sigma_p\setminus J}$. Breuil's conjecture. (1) For a character $\chi:T_p\rightarrow E^{\times}$, $(\mathfrak m_\rho,\chi)\in\mathcal E(U^p)_{\overline\rho}$ if and only if there exist $s\in\mathcal S_2^{|S_p|}$ and $J\subseteq\Sigma(z_s)_{\widetilde v}$ such that $\chi=(\delta_s)_J^c$. (2) For $s\in\mathcal S_2^{|S_p|}$ and $J\subseteq\Sigma(z_s)$, the point $(z_s)_J^c:=(\mathfrak m_\rho,(\delta_s)_J^c)$ lies moreover in $\mathcal E(U^p,\underline\lambda_{\Sigma_p\setminus J})_{\overline\rho}$. This is the eigenvariety formulation equivalent to the locally analytic socle statement and predicts all companion points together with their expected classicality weights. The source supplies no resolution evidence for this formulation.