Smoothness of Lyapunov-function homotopies

Ascertain whether the homotopy between Lyapunov functions that preserves global asymptotic stability (from a smooth proper Lyapunov function to the quadratic function) can be chosen to consist entirely of smooth functions throughout the homotopy.

Background

The construction in Proposition prop:GAS:homotopy:semiflow produces a homotopy of Lyapunov functions that is continuous but may fail to be smooth due to the use of homeomorphisms in Step (iii) of the proof.

The authors explicitly state that whether smoothness can be maintained along the entire homotopy is currently unknown.