Polynomial ramification criterion for b-ramified power series

Let $b$ be a positive integer, let $p$ be a prime, and let

$$f(X)=X+\sum_{i=1}^{\infty}a_iX^{i+b}\in\mathcal{N}(\mathbb{F}_p)$$

be of infinite order. The series is $b$-ramified when its ramification numbers satisfy $i_n(f)=b(1+p+\dots+p^n)$ for every $n\geq 0$.

Polynomial ramification criterion. For every positive integer $b$, there exists a polynomial $P_b\in\mathbb{Z}[x_1,\dots,x_{b+1}]$ such that $f$ is $b$-ramified if and only if $p\nmid b$, $a_1\neq 0$, and

$$P_b(a_1,\dots,a_{b+1})\neq 0.$$

This conjecture would give a finite criterion for the ramification of these power series using only their first $b+1$ nontrivial coefficients. The criterion is proved in the paper for $b=3$ through an explicit polynomial condition, while the asserted existence and prime-independence of $P_b$ for general $b$ is supported by computations and remains open.