Faithful invariant ideals are controlled by fixed points

Let $G$ be a free abelian pro-$p$ group of finite rank, let a uniform pro-$p$ group $\Gamma$ act linearly on $G$, and let $kG$ denote the associated Iwasawa algebra. An ideal $I$ is faithful when

$$(1+I)\cap G=1.$$

Fixed-point control conjecture. If $I$ is a $\Gamma$-invariant faithful prime ideal of $kG$, then $I$ is controlled by the subgroup of $\Gamma$-fixed points $G^\Gamma$:

$$I=(I\cap kG^\Gamma)kG.$$

This conjecture is inspired by Roseblade's theorem and proposes a structural description of faithful invariant prime ideals in Iwasawa algebras. The supplied text gives supporting evidence from linear actions with sufficiently large image, but does not state a resolution.