The global-dimension criterion for Iwasawa algebras

Let $G$ be a compact $p$-adic analytic group, let $H$ be a closed normal subgroup of $G$, and write $\Omega_{G,H}$ and $\Lambda_{G,H}$ for the corresponding Iwasawa algebras. For $x\in G$, let the centralizer of $xH$ in $G$ mean the subgroup of elements of $G$ whose conjugation fixes the coset $xH$. Global-dimension conjecture. The global dimension of $\Omega_{G,H}$, and likewise that of $\Lambda_{G,H}$, is infinite if and only if there exists an element $x\in G$ of order $p$ such that the centralizer of $xH$ in $G$ is open in $G$. This would make the condition in Theorem J(iv), previously shown to be sufficient, also necessary and sufficient; the surrounding discussion gives examples showing that the absence of elements of order $p$ is not necessary for finite global dimension.