Gevrey regularity conjecture for the parametrization of invariant circles

Let $u_{\varepsilon}$ denote the parametrization of the quasi-periodic invariant circle for the dissipative standard map, viewed as a function of the complex parameter $\varepsilon$. Gevrey-regularity conjecture. The parametrization $u_{\varepsilon}$ belongs, as a function of $\varepsilon$, to a Gevrey class $G^\sigma$ whose index $\sigma$ is close to $1$. The formal series is not analytic in a neighborhood of $\varepsilon=0$, because no ball around the origin has a convergent formal power series, but the numerical coefficient growth suggests regularity very close to the analytic class; the precise Gevrey index remains unspecified.