The p-adic Hilbert–Smith conjecture

For a prime $p$, let $\widehat{\mathbb{Z}}_p$ denote the $p$-adic group. The p-adic Hilbert–Smith conjecture. For every prime $p$, there is no faithful action of $\widehat{\mathbb{Z}}_p$ on a topological manifold. This is the $p$-adic formulation equivalent to the Hilbert–Smith conjecture in the structural reduction described by the source. The source says that this formulation has been proved in different contexts, including for topological manifolds of dimension at most $3$, but does not state a full resolution in all dimensions.