Socle conjecture for refinements of automorphic Galois representations

Let $\mathcal{R}$ be a refinement, let $\lambda$ be the algebraic weight appearing in the associated locally analytic representation, and let $w_0$ be the longest Weyl-group element. For $w\in\prod_{v\in S_p}\mathcal{S}_n^{[F_{\tilde v}:\mathbb{Q}_p]}$, let $w_{\mathcal{R}}$ be the tuple of permutations associated with the trianguline points attached to the refinement, and let $\preceq$ be the Bruhat order. Socle conjecture.

$$\operatorname{Hom}_{G_p}\Big(\mathcal{F}_{\overline B_p}^{G_p}\big(\overline L(-ww_0\cdot\lambda)^\vee,\underline{\delta}_{\mathcal{R},\mathrm{sm}}\delta_{B_p}^{-1}\big),\widehat S(U^p,L)_{\mathfrak{m}^S}^{\mathrm{an}}[\mathfrak{m}_{\rho}]\Big)\ne0$$

if and only if $w_{\mathcal{R}}\preceq w$. This generalizes the socle conjecture cited in the paper and predicts precisely which locally analytic constituents occur in completed cohomology for a given refinement; the supplied passage does not state a resolution.