$\DeclareMathOperator\GL{GL}$The question is about the paper: Caraiani, Emerton, Gee, Geraghty, Paskunas, and Shin - Patching and the $p$-adic local Langlands correspondence. Let $F$ be a finite extension of $\mathbb{Q}_p$ and let $E$ be another finite extension of $\mathbb{Q}_p$, which is assumed to be sufficiently large. To a Galois representation $r : \mathrm{Gal}(\overline{F}/F) \to \GL_n(E)$ the authors associate $V(r)$, which is a continuous, unitary admissible $E$-Banach space representation of $\GL_n(F)$. Assume that $r$ is potentially semi-stable, not necessarily generic, and it lies on an automorphic component. By Proposition 4.33 of this paper, the locally algebraic vectors of $V(r)$ are $V(r)^{\text{l.alg}} \simeq \pi \otimes \pi_{\text{alg}}(r)$, where $\pi$ is smooth and admissible.
Question : How to prove that $\pi$ as above has an irreducible generic sub-representation? A representation theoretic argument would be preferred, but any other suggestions are welcome.
