The dimension bound conjecture for infinite-dimensional completed-cohomology modules

Let $K=\mathrm{SL}_n(\mathbf{Z}_p)$ and let $K(p)$ be its principal congruence subgroup. Write $\Lambda=\mathbf{Z}_p[[K(p)]]$. For a finitely generated $\Lambda$-module $M$, let its module dimension be the dimension defined by the relevant Iwasawa-theoretic growth theory. Dimension bound conjecture. If $M$ is infinite-dimensional over $\mathbf{F}$, then the dimension of $M$ is at least $n-1$. This is the natural analogue of an earlier dimension bound in the paper and is used to control completed cohomology. The supplied text does not establish the assertion or give evidence of its resolution.