Primitivity of finitely presented monomial algebras

We consider prime monomial algebras and we prove a special case of a conjecture of Jason P. Bell and Agata Smoktunowicz. We show that a prime finitely presented monomial algebra is either primitive or satisfies a polynomial identity and has GK dimension 1. More generally, we show that this result holds for automaton algebras; that is, monomial algebras whose set of nonzero words is recognized by a finite state machine.