The Ding–Gao–Hjorth conjecture on essentially hyperfinite abelian-group actions

Let $G$ be an abelian Polish group acting in a Borel manner on a Polish space $X$. Let $E^X_G$ denote the orbit equivalence relation induced by this action, and let $E\leq_B E^X_G$ mean that $E$ is Borel reducible to $E^X_G$. An equivalence relation is essentially countable if it is Borel reducible to a countable Borel equivalence relation, and essentially hyperfinite if it is Borel reducible to a hyperfinite Borel equivalence relation.

Ding–Gao–Hjorth conjecture. If $E\leq_B E^X_G$ and $E$ is essentially countable, then $E$ is essentially hyperfinite.

The conjecture would significantly improve Hjorth's $\ell^1$ dichotomy. It is supported by results for countable groups, non-Archimedean groups, and further cases treated in the paper, but remains open.