The countable invariant uniformization characterization of reducibility to countable

Let $E$ be a Borel equivalence relation on a Polish space $X$. Say that $E$ is reducible to countable if there are a Polish space $Z$, a countable Borel equivalence relation $F$ on $Z$, and a Borel function $S\colon X\to Z$ such that

$$x_1 E x_2 \iff S(x_1) F S(x_2).$$

For a Polish space $Y$, an $E$-invariant countable uniformization of an $E$-invariant Borel set $P\subseteq X\times Y$ with full projection is a Borel function $f\colon X\to Y^\mathbb{N}$ such that $(x,f(x)_n)\in P$ for all $x,n$, and $x_1 E x_2$ implies

$$\{f(x_1)_n:n\in\mathbb{N}\}=\{f(x_2)_n:n\in\mathbb{N}\}.$$

We say that $E$ satisfies measure, category, or $K_\sigma$ countable invariant uniformization when the corresponding uniformization property holds for every $Y$, measure or category structure, and $P$ in the relevant uniformization theorem, provided $P$ is $E$-invariant.

Countable invariant uniformization conjecture. Let $E$ be a Borel equivalence relation on a Polish space $X$. Then the following are equivalent: (a) $E$ is reducible to countable; (b) $E$ satisfies measure countable invariant uniformization; (c) $E$ satisfies category countable invariant uniformization; (d) $E$ satisfies $K_\sigma$ countable invariant uniformization.

The implication from reducibility to countable to each of the three uniformization properties follows from the preceding argument, while the converse equivalences are conjectured and remain unresolved.