General characterization of arbitrarily distorted diffeomorphisms

Let $M$ be a compact smooth manifold, and let $h\in\operatorname{Diff}^1(M)$. A distortion element is an element whose distortion function measures its word length in a finitely generated subgroup relative to its powers. A Lipschitz action of a finite-dimensional torus on $M$ is an action of a finite-dimensional torus by Lipschitz diffeomorphisms.

General distortion conjecture. The element $h$ is a distortion element in $\operatorname{Diff}^1(M)$ whose distortion function can be chosen to grow faster than any given function if and only if some finite power of $h$ is contained in a Lipschitz action of a finite-dimensional torus on $M$.

This conjecture proposes a characterization of those diffeomorphisms that can have arbitrarily rapid distortion in terms of torus actions. The preceding constructions establish broad examples for rotations, but the general equivalence for compact smooth manifolds remains open.