Sharpness of $C^0$ conjugacy for the non-autonomous differential equations with Lipschitzian perturbation

The classical $C^0$ linearization theorem for the non-autonomous differential equations states the existence of a $C^0$ topological conjugacy between the nonlinear system and its linear part. That is, there exists a homeomorphism (equivalent function) $H$ sending the solutions of the nonlinear system onto those of its linear part. It is proved in the previous literature that the equivalent function $H$ and its inverse $G=H^{-1}$ are both H\"{o}lder continuous if the nonlinear perturbation is Lipschitzian. Questions: is it possible to improve the regularity? Is the regularity sharp? To answer this question, we construct a counterexample to show that the equivalent function $H$ is exactly Lipschitzian, but the inverse $G=H^{-1}$ is merely H\"{o}lder continuous. Furthermore, we propose a conjecture that such regularity of the homeomorphisms is sharp (it could not be improved anymore). We prove that the conjecture is true for the systems with linear contraction. Furthermore, we present the special cases of linear perturbation, which are closely related to the spectrum.