The characteristic-$p$ analytic cohomology conjecture

Let $K$ be a nonarchimedean field of characteristic $p$. Let $\Gamma$ be a compact $p$-adic Lie group, let $H$ be a pro-nilpotent closed subgroup of $\Gamma$, and let $M$ be a Banach space over $K$ equipped with a continuous analytic action of $\Gamma$.

Characteristic-$p$ analytic cohomology conjecture. (a) If there is a sequence of subgroups $H_j$ of $H$ forming a neighborhood basis of the identity such that

$$H^i_{\operatorname{ct}}(H_j,M)=0$$

for all $i\geq 0$ and all $j$, then

$$H^i_{\operatorname{ct}}(\Gamma,M)=0$$

for all $i\geq 0$. (b) If the action of $\Gamma$ is $K$-linear and there is such a sequence with

$$\dim_K H^i_{\operatorname{ct}}(H_j,M)<\infty$$

for all $i\geq 0$ and all $j$, then

$$\dim_K H^i_{\operatorname{ct}}(\Gamma,M)<\infty$$

for all $i\geq 0$.

This conjecture proposes extending the paper's main result from procyclic to pro-nilpotent subgroups. The pro-$p$ procyclic case is covered by the main theorem when the hypotheses hold for $H$ itself; the general pro-nilpotent case remains open.