Andrade–Miller–Pratt–Trinh conjecture on infinite non-trivial $F$-sets

Throughout, let $q$ be a prime power, let $I_q$ be the set of all monic irreducible polynomials in $\mathbb{F}_q[x]$, and call a subset $A$ of $I_q$ an $F$-set if, for every $f(x)\in A$, all monic irreducible polynomials dividing $f(x)-f(0)$ also belong to $A$. An $F$-set is non-trivial if it is different from $I_q$. Andrade–Miller–Pratt–Trinh conjecture. For every prime power $q$, there exist an infinite, non-trivial $F$-set. The paper proves this conjecture for every fixed $q$, so the claim is solved.