Suppose $E,F$ are Borel equivalence relations on Polish spaces $X,Y$, respectively. Under strong enough determinacy axioms, is it true that $E$ Borel reduces to $F$ iff there is an injective map from the quotient set $X/E$ to $Y/F$? Is it at least true for some class of equivalence relations? This answer mentions that:
In natural situations, in particular in models of determinacy, we can replace "Borel reducibility", that is, "Borel cardinality" via Borel injections by actual cardinality.
so I guess the answer to my question is yes. But the best I can do myself is this: suppose there is an injective set mapping $f:X/E\rightarrow Y/F$, and consider $\Gamma=\{(x,y)\in X\times Y:f([x])=[y]\}$. Under strong determinacy all sets can be uniformized, so we have a reduction $g:X\rightarrow Y$, which is at least Baire measurable, and therefore Borel measurable on a comeager set. Can this be improved further?
Edit: I realized that basically I'm looking for some $E$ that is Baire reducible but not Borel reducible to $F$. We want to find such $E,F$ under $\mathsf{AD}_\mathbb{R}$ instead of $\mathsf{ZFC}$; see the comments below.
