Complementation of deterministic Streett automata

There are a lot of work about bounds on the complementation of Streett automata (see e.g. this paper). They talk about the general setting of nondeterministic Streett automata. But what about deterministic ones?

Streett automata are closed under determinisation although with an exponential blow-up. And complementation has an exponential complexity. So I would suppose that complementation of deterministic Streett automata would be simpler, maybe polynomial?

So, how do I complement deterministic Streett automata? Is the task computationally easy?

Edit: as far as I can tell, this question is equivalent to asking how to translate a deterministic Rabin automaton to an equivalent deterministic Streett automaton.

This paper gives an example of a deterministic Rabin automaton with $\mathcal{O}(n)$ states such that there is no nondeterministic (hence no deterministic) equivalent Streett automaton with less than $2^n$ states.

I guess there is no hope for a polynomial complementation, even if the Streett automaton is deterministic.

StackExchange Computer Science Q#155074, answer score: 4