Existence of nontrivial infinite F-sets over finite fields

Let $\mathbb{F}_q$ be a finite field. An $F$-set is a set $\mathcal{S}\subset\mathbb{F}_q[x]$ such that, whenever $P\in\mathcal{S}$ is monic and irreducible with constant term $\alpha_P$, every monic irreducible polynomial $Q$ dividing $P-\alpha_P$ also belongs to $\mathcal{S}$. Existence conjecture. For any finite field $\mathbb{F}_q$, there exists a nontrivial, infinite $F$-set. The paper proves this for $\mathbb{F}_p$ when $p$ is prime and $p\equiv 2$ or $5\pmod 9$; the conjecture asks for the result over every finite field, extending beyond the cases established in the paper.