Partial classicality conjecture for companion points

Let $\rho$, $\Sigma_p$, $C(\rho)$, $\Sigma(\rho)$, $\mathcal E(U^p)_{\overline\rho}$, and $\delta_J^c$ be as in the preceding setup. Partial classicality conjecture. For every subset $J$ satisfying

$$\Sigma_p\setminus C(\rho)\subseteq J\subseteq\Sigma(\rho),$$

the point $(\mathfrak m_\rho,\delta_J^c)$ lies in

$$\mathcal E\left(U^p,\operatorname{wt}(\delta)_{\Sigma_p\setminus J}\right)_{\overline\rho}.$$

This is identified in the source as the unproved part of the preceding locally analytic socle conjecture: the converse socle implication would follow from this eigenvariety classicality assertion.