Is there a Borel equivalence relation $E$ on $[\omega]^\omega$ such that $E \not \leq_B E_0$ and for any $a \in [\omega]^\omega$ we have that $E \upharpoonright [a]^\omega$ is Borel bireducible with $E$?
I have a feeling I have seen this question (possibly with an answer) somewhere, but I cannot find it now. In case this really can be found somewhere, I would appreciate the reference.
In case the answer is yes, is it still yes if we replace $E_0$ by something more complex?
Note that by Theorem 8.17 (Mathias) from Canonical Ramsey theory on Polish spaces by Kanovei, Sabok and Zapletal, we have that for a countable Borel equivalence relation $E$, there will be some $a \in [\omega]^\omega$ with $E \upharpoonright [a]^\omega \leq_B E_0$, so we can exclude countable $E$.
