Co-finiteness of VASS coverability languages

We study the class of languages recognized by multi-counter finite state automata. These are finite automata reading letters from a finite alphabet A, equipped with n counters of natural numbers, which can be incremented or decremented by transitions. The acceptance condition requires the last state to be from the final set of states. This is equivalent to the language acceptors associated with coverability problems for labelled Petri Nets or labelled Vector Addition Systems with States (VASS).We show that the problem of whether the complement of the language has finitely many words (i.e., whether it is «almost equal» to A*) is decidable. We do this by a reduction to the universality problem (i.e., whether it is «equal» to A*)