Finite-length conjecture for locally analytic vectors

Let $G$ be a ${\mathbb Q}_p$-analytic group over $L$, and let $\Pi$ be an absolutely irreducible admissible $L$-Banach space representation of $G$. Denote by $\Pi_{\rm an}$ the space of locally analytic vectors in $\Pi$. Finite-length conjecture. The space $\Pi_{\rm an}$ is a $G$-representation, topologically of finite length, and its continuous $G$-equivariant endomorphisms are all scalar:

$$\operatorname{End}_{L[G]}(\Pi_{\rm an})=L.$$

In particular, $\Pi_{\rm an}$ has an infinitesimal character. This conjecture is presented as a deep open problem in the theory of $p$-adic representations of $p$-adic Lie groups; the surrounding discussion gives no resolution.