Stability preservation of the Sontag–Coron homotopy

Prove or disprove that the homotopy of vector fields H(s,x) defined by H(0,x)=X(x), H(1,x)=-x, and H(s,x)=(1/s)(varphi^{s/(1-s)}(x;X)-x) for s in (0,1), where varphi(·;X) is the flow of X and the origin is globally asymptotically stable under X, preserves global asymptotic stability of the origin for all s in [0,1].

Background

Equation (equ:flow:homotopy:sontag:coron) defines a classical homotopy (used by Sontag and Coron) from a given vector field X to the canonical linear vector field −x via time-τ maps of X. While it is known not to introduce new equilibria, it is not established in general whether it preserves global asymptotic stability (GAS) along the path.

The scalar and linear cases preserve GAS, but the general nonlinear case remains unresolved.