Gevrey regularity conjecture for the Lindstedt series of the dissipative standard map

Let $u$ be a parameter and let $\boldsymbol{\mathcal D}( u,1)$ denote the corresponding set of frequencies. For $\omega\in\mathcal D(\nu,1)$, consider the Lindstedt series

$$u_\varepsilon=\sum_k u_k\varepsilon^k$$

of quasi-periodic orbits for the dissipative standard map, and let $\|\cdot\|_\rho$ denote the norm at analyticity width $\rho$. Gevrey regularity conjecture. The Lindstedt series belongs to a Gevrey class with Gevrey exponent $\sigma\leq 0.307$; specifically,

$$\|u_n\|_\rho\leq CR^n n^{\sigma n},\qquad \sigma\leq 0.307,$$

with $\rho\leq 10^{-7}$. This is a reformulation of Conjecture 9 from the cited earlier work for a more general range of frequencies, based on numerical computations and fits; its resolution is not established in the supplied text.