Decompletion conjecture for compact p-adic Lie groups

Let $G$ be a compact $p$-adic Lie group, and let $M$ and the associated period rings and locally analytic-vector functor be as in Theorem 1. The Lie algebra $\mathrm{Lie}(G)$ is not assumed to be abelian. Decompletion conjecture. The statement of Theorem 1 holds for any compact $p$-adic Lie group $G$ (with no assumption on $\mathrm{Lie}(G)$). This conjecture proposes that the decompletion result extends beyond the Lie-algebra hypothesis used in the theorem.