Floquet stability conjecture for the reduced periodic orbit

Let $u$ be the reduced dynamical system

$$\dot{x}=f_x(x,h_y(x),0),$$

associated with the planar dynamics in the paper, and let $\gamma_0$ be the periodic orbit whose existence is established by Lemma. Floquet stability conjecture. For $\sigma=4$, the orbit $\gamma_0$ is asymptotically stable and has Floquet multipliers

$$\rho_1=1,\qquad \rho_2<1.$$

If true, this verifies the stability hypothesis needed to apply Fenichel theory and deduce persistence of a stable limit cycle for the perturbed system at sufficiently small positive perturbation parameters. The supplied text gives no resolution of this conjecture.