The coefficient congruence conjecture for Böttcher coordinates of a perturbed monomial

Let $p$ be a prime and let

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

Write the Böttcher coordinate of $\varphi$ as

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

where the coefficients $a_k$ are integers. Coefficient congruence conjecture. The following congruences hold:

$$k\equiv0\pmod p\quad\Longrightarrow\quad a_k\equiv(-1)^{k/p}\pmod p,$$ $$k\equiv-2\pmod p\quad\Longrightarrow\quad a_k\equiv-1\pmod p,$$ $$k\equiv-1\pmod p\quad\Longleftrightarrow\quad a_k\equiv0\pmod p,$$

with the one exception that, for $p=2$, $a_1\equiv1\pmod2$. This conjecture is based on computations for primes $2\le p\le11$ and $0\le k\le50$; its general validity remains open.