Maximal-order conjecture for completed group algebras of dihedral-free groups

Let $G$ be a compact $p$-adic analytic group. Write $\Omega_G$ for its completed group algebra over the coefficient ring used in the paper. Call $G$ dihedral-free if every orbital closed subgroup $H$ of dimension $1$ is isomorphic to $\mathbb{Z}_p$, where orbital means that $H$ has finitely many $G$-conjugates. Assume that $\Omega_G$ is prime. Maximal-order conjecture. $\Omega_G$ is a maximal order if and only if $G$ is dihedral-free. The conjecture proposes the group-theoretic criterion governing when the prime Iwasawa algebra $\Omega_G$ is a maximal order; the surrounding discussion presents it as the conjectured answer to the preceding question and mentions evidence in its support, but no resolution is supplied here.