  Suppose $R$ is a profinite ring. We construct a large class of profinite
groups $\widehat{{\scriptstyle\bf L}'{\scriptstyle\bf H}_R}\mathfrak{F}$,
including all soluble profinite groups and profinite groups of finite
cohomological dimension over $R$. We show that, if $G \in
\widehat{{\scriptstyle\bf L}'{\scriptstyle\bf H}_R}\mathfrak{F}$ is of type
$\operatorname{FP}_\infty$ over $R$, then there is some $n$ such that
$H_R^n(G,R [[ G ]]) \neq 0$, and deduce that torsion-free soluble pro-$p$
groups of type $\operatorname{FP}_\infty$ over $\mathbb{Z}_p$ have finite rank,
thus answering the torsion-free case of a conjecture of Kropholler.
