Is there a polynomial time algorithm to tell if an NFA over an unary alphabet is universal?

Given an Nondeterministic Finite State Automaton with $n$ states over an unary alphabet, is there some algorithm to check if the automata is universal in time polynomial in the number of states?

I would not expect this would work over an alphabet of size two or more, since it is PSPACE-complete to check whether a regular expression is universal. However, since regular languages over unary alphabets have nice characterizations, I would expect there would be some way to check the universality of an NFA over an unary alphabet.

Determining whether a unary NFA is universal is coNP-complete, as proved by Stockmeyer and Meyer, Word Problems Requiring Exponential Time. See Gruber and Holzer, Computational Complexity of NFA Minimization for
Finite and Unary Languages, for more results in this vein.

StackExchange Computer Science Q#97746, answer score: 4