On the existence of infinite, non-trivial F-sets

In this paper we prove a conjecture of J. Andrade, S.J. Miller, K. Pratt and M. Trinh, showing the existence of a non-trivial infinite F  -set over Fq[x]Fq[x] for every fixed q. We also provide the proof of a refinement of the conjecture, involving the notion of width of an F-set, which is a natural number encoding the complexity of the set.