The Gap Conjecture for finite group actions on 3-manifolds

Let $F$ be a finite group acting on a $3$-manifold by homeomorphisms. The modulus of continuity of an element is denoted by $\operatorname{moc}$. An action has subexponential moc when the moc of each element grows subexponentially.

Gap Conjecture. If the moc of each element of $F$ is subexponential, then the action is topologically conjugate to a smooth ($C^\infty$) action.

The conjecture predicts a gap in dimension three between isometric actions, with no stretching after choosing a suitable metric, and actions requiring substantial stretching. It would imply a fully exponential lower bound for the inherent modulus of continuity of the Bing involution, but no resolution is supplied here.