The one-based/field-like dichotomy conjecture for full uniformly powerful pro-p groups

Let $\mathcal{G}=(G,I)$ be a uniformly powerful pro-$p$ group that is full as a profinite group. For each $n$, let $P_n(G)$ denote the corresponding term in the lower $p$-series, and regard the finite quotient $G/P_n(G)$ as an $L_{\operatorname{prof}}$-structure, with each $K_i$ interpreted by $K_iP_n(G)/P_n(G)$. One-based/field-like dichotomy conjecture. The following conditions are equivalent:

1. The ring $\mathbb{Z}_p$ is not interpretable in $\mathcal{G}$.
2. For every sentence $\sigma$ in the language $L_{\operatorname{prof}}$, there is $N\in\omega$ such that either every quotient $(G/P_n(G),I)$ satisfies $\sigma$ for $n>N$, or every such quotient satisfies $\neg\sigma$.
3. The group $G$ is nilpotent-by-finite.

This conjecture proposes a model-theoretic one-based/field-like dichotomy for compact $p$-adic analytic groups: failure of interpretation of the $p$-adic integers is predicted to coincide with eventual elementary stabilization of the finite quotients and with nilpotence up to finite index.