Small-homeomorphism compact-subgroup conjecture

Let $M$ be a connected $n$-manifold with metric $d$, and let $U\subseteq M$ be open. For $\epsilon>0$, consider the subset

$$\left\{\phi\in\operatorname{Homeo}(M)\bigm|d(x,\phi(x))<\epsilon\text{ for all }x\in U\right\}.$$

Small-homeomorphism compact-subgroup conjecture. There exists $\epsilon>0$ such that this subset of $\operatorname{Homeo}(M)$ contains no nontrivial compact subgroup. For any specific manifold $M$, this is presented as a reformulation of the Hilbert–Smith conjecture. Its status therefore follows the status of that reformulation, but no separate resolution is supplied for the general statement.