Locally analytic de Rham criterion for completed cohomology representations

Let $L$ be the local field appearing in the construction, let $E$ be a finite extension of $L$ containing all embeddings of $L$ into $\overline{\mathbb{Q}}_p$, and let

$$\rho:\operatorname{Gal}_F\longrightarrow \operatorname{GL}_2(E)$$

be a two-dimensional continuous absolutely irreducible representation. Let $\Pi(\rho)$ be the associated unitary Banach representation of $\operatorname{GL}_2(L)$, and assume $\Pi(\rho)\neq 0$. For a subset $J$ of the embeddings of $L$ into $E$, write $J^c$ for its complement, and let $\Pi(\rho)^{J^c\textup{-la},J\textup{-lalg}}$ denote the corresponding partially locally analytic, partially locally algebraic vectors. Locally analytic de Rham criterion. For every such subset $J$,

$$\Pi(\rho)^{J^c\textup{-la},J\textup{-lalg}}\neq 0$$

if and only if $\rho$ is $J$-de Rham with distinct $J$-Hodge--Tate weights. This conjecture is intended to characterize the de Rham properties of $\rho$ through locally analytic vectors in completed cohomology; it refines the known implication from appropriate locally algebraic vectors to de Rham representations and would give a criterion for partial de Rhamness in every set of embeddings.