Fine-grained complexity of omega-automata inclusion

The inclusion problem asks whether the language of an automaton $A$ is included in that of an automaton $B$. If the automaton $B$ is deterministic, this problem can be solved in almost quadratic time by a product construction, and this bound is optimal: a $o(n^2)$ algorithm would contradict ETH (Wehar 2016). For automata over infinite words, the problem is still in PTIME (in NL in fact), but its exact complexity is unknown. $**Goal:**$ Find upper and lower bounds on $k$ such that the inclusion of deterministic parity automata can(not) be checked in time $O(n^k)$.