The dimension bound conjecture for smooth admissible representations of general linear groups

Let $\pi$ be an irreducible infinite-dimensional smooth admissible representation of $\mathrm{GL}_n(\mathbf{Q}_p)$, and let $\pi^{\vee}=\operatorname{Hom}(\pi,\mathbf{F}_p)$. Let $\Lambda_{\mathbf{F}_p}$ denote the completed group ring used in the paper. Representation-theoretic dimension bound conjecture. The dimension of $\pi^{\vee}$ as a $\Lambda_{\mathbf{F}_p}$-module is at least $n-1$. This is presented as a representation-theoretic substitute for the preceding module-theoretic conjecture and is intended to imply stability results for completed cohomology. The supplied text gives no proof or resolution of the assertion.