Infinitely many prime divisors for non-geometric generalized derangement sequences

Let $\mathbb{N}$, $\mathbb{P}$ and $\mathbb{Z}$ denote the non-negative integers, prime numbers and integers, respectively. For $f,g,h\in\mathbb{Z}[X]$, let $\mathbf{a}(f,g,h)=(a_n)_{n\in\mathbb{N}}\in\mathcal{R}$ be defined by

$$a_0=g(0),\qquad a_n=f(n)a_{n-1}+g(n)\left(h(n)\right)^n\quad(n>0).$$

For a sequence $\mathbf{a}=(a_n)_{n\in\mathbb{N}}$, define

$$\mathcal{P}_{\mathbf a}=\left\{p\in\mathbb{P}:\exists_{n\in\mathbb{N}}\quad p\mid a_n\right\}.$$

The non-geometric prime-divisor conjecture. If $\mathbf{a}\in\mathcal{R}$ is not a geometric progression, then the set $\mathcal{P}_{\mathbf a}$ is infinite.

The conjecture asks whether every generalized derangement sequence that is not geometric has infinitely many prime divisors among its terms. The paper states that this conjecture is false, so the claim is refuted.