Periodic correction conjecture for Lindstedt-series coefficient growth

Let $u_k$ be the coefficients of the Lindstedt series and define, for $\rho=10^{-7}$, the normalized logarithmic growth

$$A_\rho(k)=\frac{1}{k}\log\|u_k\|_\rho.$$

Periodic correction conjecture. For $k\gg 1$, $A_\rho(k)$ approximately satisfies

$$A_\rho(k)\approx \log(R)+\sigma\log(k)+k^{-\beta}f(k),$$

where $\beta\approx 1$ and $f(k)$ is a periodic function of period $3$. The conjectured asymptotic describes a period-three oscillatory correction whose amplitude decays like $k^{-1}$.