Dimension-5 diffeomorphism regularity in the Grüne–Sontag–Wirth construction

Determine whether, in dimension n=5, the homeomorphism T that transforms a smooth Lyapunov function V into a radial form V(T^{-1}(x))=γ(∥x∥) can be chosen to be a diffeomorphism on R^n\{0}.

Background

The proof relies on a result that for n≠5 the conjugating map T can be taken diffeomorphic on R^n{0}, which enables defining a corresponding vector field and Lyapunov pair.

The status for n=5 remains unsettled; resolving this would impact the extension of the main homotopy result and relates to deep questions in differential topology.