Locally analytic socle conjecture for ${\rm GL}_2(L)$

Let $S\subseteq\Sigma_L$, let $I^c_S(\alpha,\mathbf h)=C(1,s_J)$ be the corresponding locally analytic representation, and let $\widehat S(U^{\mathfrak p},E)_{\overline\rho}^{\mathrm{an}}[\mathfrak m_\rho]$ be the localized locally analytic automorphic representation. Locally analytic socle conjecture. Assuming Hypothesis,

$$I^c_S(\alpha,\mathbf h)\hookrightarrow\widehat S(U^{\mathfrak p},E)_{\overline\rho}^{\mathrm{an}}[\mathfrak m_\rho]$$

if and only if $S\subseteq\Sigma(y)$. This is the locally analytic socle prediction for $\mathrm{GL}_2(L)$ and is equivalent in the paper to the companion-point formulation.