Recall that an infinite pro-$p$ group $G$ is called just-infinite if all non-trivial closed normal subgroup of $G$ have finite index.
Question: Let $G$ be an insoluble $p$-adic analytic just-infinite pro-$p$ group. Is it true that $G$ is torsion-free?
If $\text{dim}~G<p-1$, then it is true, cf. Theorem 1.3 in [Klopsch, Benjamin. "On the Lie theory of p-adic analytic groups." Mathematische Zeitschrift 249 (2005): 713-730.] Any references or counterexample will be appreciated.
Added. Let $p$ be an odd prime. The following are equivalent:
(1) Any insoluble $p$-adic analytic just-infinite pro-$p$ group is torsion-free.
(2) Any Sylow-pro-$p$ subgroup of ${\rm SL}_n(\mathbf{Z}_p)$ is torsion-free for any $n\geq 2$.
Sketch of proof:
$(1)\implies (2)$: Note that any Sylow-pro-$p$ subgroup of ${\rm SL}_n(\mathbf{Z}_p)$ is an insoluble $p$-adic analytic just-infinite pro-$p$ group;
$(2)\implies (1)$: Let $G$ be any insoluble $p$-adic analytic just-infinite pro-$p$ group. Then $G$ is isomorphic to a closed subgroup of ${\rm GL}_n(\mathbf{Z}_p)$ for some positive integer $n$. Since the abelianization of $G$ is a finite $p$-group, $G$ is isomorphic to a closed subgroup of ${\rm SL}_n(\mathbf{Z}_p)$. Thus, $G$ must be torsion-free.