Which Finitely Ambiguous Automata Recognize Finitely Sequential Functions?

Weighted automata, especially min-plus automata that operate over the tropical semiring, have both a beautiful theory and important practical applications. In particular, if one could find a sequential or finitely sequential equivalent to a given (or learned) min-plus automaton, one could increase performance in several applications. But this question has long remained open even as a decision problem. We show that existence of a finitely sequential equivalent for a given finitely ambiguous min-plus automaton is decidable.