The increasing union problem for CBERs

A CBER, short for countable Borel equivalence relation is an equivalence relation $E$ on a Polish space $X$ such that $E$ is a Borel subset of $X^2$ and each equivalence class of $E$ is countable.

A CBER $E$ on $X$ is hyperfinite if there are CBERs $E_0 \subseteq E_1 \subseteq E_2 \subseteq \ldots$ such that $E = \bigcup_{n \in \N}E_n$ and for each $n$, all of $E_n$'s equivalence classes are finite.

Sometimes a CBER which is an increasing union of countably many hyperfinite CBERs is called hyperhyperfinite. Hence the question can be restated as asking whether every hyperhyperfinite CBER is hyperfinite.

Conley, Jackson, Marks, Seward and Tucker-Drob [CJMSTD23] showed that if a CBER $E$ is a generated by a graph which can be written as a countable increasing union of graphs of finite Borel asymptotic dimension (a stronger assumption than hyperfiniteness) then $E$ is hyperfinite.

This problem is connected to several other well-known problems about hyperfinite CBERs. For example, one such question is whether the collection of codes for hyperfinite CBERs is $\Sigma^1_2$-complete. Frisch, Shinko and Vidnyánszky [FSV24] have shown that if there is a hyperhyperfinite CBER which is not hyperfinite then this set of codes is $\Sigma^1_2$-complete.