Equivalence of hyper-hyperfinite and hyperfinite-over-hyperfinite classes

Determine whether the class of hyper-hyperfinite Borel equivalence relations coincides with the class of hyperfinite-over-hyperfinite Borel equivalence relations.

Background

The paper discusses two central generalizations of hyperfiniteness: hyper-hyperfinite relations (increasing unions of hyperfinite relations) and hyperfinite-over-hyperfinite relations (admitting Borel Z^2-orderings on classes). Both are shown to be Fréchet-amenable, suggesting a unified perspective.

Despite these connections, it remains unsettled whether these two classes are actually the same, i.e., whether each implies the other.