Attracting limit cycle converging to the singular cycle

Let $\alpha$ and $\xi$ be parameters with $\alpha>\xi$, let $\varepsilon>0$ be sufficiently small, and let $\Gamma_0$ denote the singular cycle described by the concatenation of the fast and slow orbit segments in the singular limit. Limit-cycle conjecture. For $0<\varepsilon\ll1$, there exists an attracting limit cycle $\Gamma_{\varepsilon}$ that converges to $\Gamma_0$ as $\varepsilon\to0$. This conjecture concerns the persistence of the singular cycle under perturbation and predicts an attracting periodic orbit for sufficiently small positive $\varepsilon$; the supplied text does not state whether it has been proved or disproved.