Middle European Mathematical Olympiad 2025 Problem T-8

Determine whether the following statement is true for every polynomial $P$ of degree at least 2 with nonnegative integer coefficients:

There exists a positive integer $m$ such that for infinitely many positive integers $n$ the number $P^n(m)$ has more than $n$ distinct positive divisors.

Remark. Here $P^n$ denotes $P$ applied $n$ times, this means $P^n(x) = \underbrace{P(P(\ldots P(x)\ldots))}_{n \text{ times}}$.