  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 $\mathbb
F_q[x]$ 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.
