Let $E/\mathbb{Q}_p$ be a finite extension. Let $\rho$ be a continuous irreducible representation of $Gal(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$ on a two dimensional $E$-vector space. To $\rho$, via the p-adic Langlands correspondence, there is an associated $E$-Banach vector space representation of $GL_2(\mathbb{Q}_p)$, which we denote by $\Pi(\rho)$.

Is it known if this construction behaves well in $p$-adic families? More precisely, if $S$ is a reduced affinoid algebra and $\rho_S$ is a continuous $S$-linear representation of $Gal(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$ on a free rank two $S$-module is there an associated $S$-Banach module, $\Pi(\rho_S)$, equipped with a contiuous $S$-linear action of $GL_2(\mathbb{Q}_p)$, such that for all $x \in Sp(S)$, $S/x \otimes\Pi(\rho_S) \cong \Pi(\rho_{S,x})$?