Exact complexity of deciding history-determinism for general parity automata

Determine the exact complexity class of the decision problem: given a nondeterministic parity automaton whose parity index is not fixed (i.e., the number of priorities is part of the input), decide whether the automaton is history-deterministic.

Background

The paper proves the 2-Token Theorem, yielding polynomial-time algorithms for fixed parity index and a PSPACE upper bound when the index is part of the input. Prior work shows NP-hardness for the general case without a fixed index. Thus, there is a gap between the NP-hard lower bound and the PSPACE upper bound that remains to be closed.

Closing this gap requires pinpointing the exact complexity of deciding whether a nondeterministic parity automaton with an unfixed parity index is history-deterministic.