Breuil's socle conjecture for locally analytic representations of $\mathrm{GL}_n(\mathbb{Q}_p)$

Let $\rho$ be an $n$-dimensional continuous representation of $\operatorname{Gal}(\overline{F}/F)$ over $E$, with $\rho_{\tilde v}$ crystalline and very regular for every $\tilde v\mid p$. For each $\tilde u$, let $w_{\tilde u}(\underline\phi)$ be a refinement and let $w_{\tilde u}^{\operatorname{alg}}(w_{\tilde u})\in S_n$ be the associated algebraic Weyl-group element. Let $C(w_{\tilde u}^{\operatorname{alg}},w_{\tilde u})$ be the corresponding locally analytic representation, and suppose $\widehat{\Pi}(\rho)_{\operatorname{lalg}}\neq 0$. Breuil's socle conjecture. The space

$$\operatorname{Hom}_{\operatorname{GL}_n(\mathbb{Q}_p)}\bigl(C(w_{\tilde u}^{\operatorname{alg}},w_{\tilde u}),\widehat{\Pi}(\rho)_{\operatorname{an}}\bigr)$$

is nonzero if and only if $w_{\tilde u}^{\operatorname{alg}}\leq w_{\tilde u}^{\operatorname{alg}}(w_{\tilde u})$, where $\operatorname{GL}_n(\mathbb{Q}_p)$ acts on $\widehat{\Pi}(\rho)$ via $i_{G,\tilde u}$. This is the global higher-rank formulation of the same expected socle description, determining which locally analytic representations occur in the completed representation; the supplied text gives no evidence that it has been resolved.