Kohen's first-zero bound conjecture for univariate constant term sequences

Let $P$ be a univariate Laurent polynomial and let $p$ be prime. For the constant term sequence $A_p(n)=\operatorname{ct}[P^n]\bmod p$, suppose there exists some $n\in\mathbb{N}$ such that $p\mid\operatorname{ct}[P^n]$. Kohen's first-zero bound conjecture. The bound in Proposition 6 holds with

$$B_{P,p}=p^{\deg(P)},$$

so there exists an $n_0<B_{P,p}$ such that $p\mid\operatorname{ct}[P^{n_0}]$. This conjecture sharpens the general automaton-theoretic bound for the location of the first zero; it was suggested by computational experiments in the univariate case, and no proof or counterexample is given here.