The asymptotic valuation and radius conjecture for perturbed monomial Böttcher coordinates

Let $p$ be a prime and let $r\ge0$. Define

$$\varphi(x)=x^{p^2}+p^{r+2}x^{p^2+1},$$

and let

$$f_\varphi(x)=x\sum_{k=0}^{\infty}\frac{a_k}{k!}x^k$$

be the associated Böttcher coordinate. Asymptotic valuation and radius conjecture. For $k\equiv0\pmod p$,

$$\operatorname{ord}_p(a_k)=\frac{1-p^{-r}}{p-1}k+O(1)\quad\text{as }k\to\infty.$$

Moreover, the radius of convergence is

$$\rho(f_\varphi):=\liminf_{k\to\infty}\left\|\frac{a_k}{k!}\right\|_p^{-1/k}=p^{-p^{-r}/(p-1)}.$$

These assertions are based on numerical experiments describing the increasing integrality of the coefficients as $r$ grows; the conjectured asymptotic valuation and exact radius remain open.