Lyapunov instability transfer conjecture for boundary equilibria in Filippov systems

Let $f$ be a Filippov system with boundary equilibrium $x^*={\bf 0}$, and let $F$ be its reduced system. Assume the setting and hypotheses of the paper, including $c_1<0$. Lyapunov instability transfer conjecture. If the equilibrium $x={\bf 0}$ is unstable, that is, not Lyapunov stable, for $F$ in, then it is also unstable for $f$ in. The conjecture would complement the proved implication that asymptotic stability of the reduced system implies asymptotic stability of the original Filippov system; the corresponding implication for asymptotic instability is false, while the Lyapunov-stability version remains to be established.