The standard-prime ideal conjecture for solvable Iwasawa algebras

Let $p$ be a prime, let $K$ be a finite extension of $\mathbb{Q}_p$, and let $G$ be a solvable uniform pro-$p$ group. Write $KG$ for the Iwasawa algebra of continuous $K$-distributions on $G$. A prime ideal $P$ of $KG$ is standard if there exists a closed, normal subgroup $H$ of $G$ such that

$$H-1\subseteq P,$$

and, setting $G':=G/H$, the ideal $\widetilde{P}:=P/(H-1)KG$ of $KG/(H-1)KG\cong KG'$ is centrally generated.

Standard-prime ideal conjecture. If $P$ is a prime ideal of $KG$, then $P$ is standard.

Standard prime ideals have particularly strong properties: in particular, they are generally completely prime, so $KG/P$ is a domain. The conjecture proposes a predictable classification of prime ideals in solvable Iwasawa algebras and addresses the broader classification problem for prime ideals in $KG$; its resolution is not established by the supplied text.