Canonical extension conjecture for the characteristic map

The characteristic map is a homeomorphism on the invariant set, with values in a cylinder. Canonical extension conjecture. The map $\widetilde{T}$ can be canonically extended to a homeomorphism

$$\widetilde{T}:\mathbb{T}^{1}\times\mathbb{R}\rightarrow\mathbb{T}^{1}\times\mathbb{R},$$

diffeomorphic to an area-preserving twist diffeomorphism $f_{V,F}$. This predicts that the characteristic map has the global structure of an area-preserving twist map, linking the driven generalized elastic chain to the dynamics of twist diffeomorphisms; the source presents this as a conjecture and gives no resolution.