The conjecture that essentially countable subrelations of abelian Polish-group orbit relations are essentially hyperfinite

Let $G$ be an abelian Polish group, and let $F$ be an orbit equivalence relation induced by a Borel action of $G$ on a Polish space. Let $E$ be a Borel equivalence relation, and write $E\leq_B F$ when $E$ is Borel reducible to $F$. 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.

Essential hyperfiniteness conjecture. If

$$E\leq_B F$$

and $E$ is essentially countable, then $E$ is essentially hyperfinite.

The statement generalizes the paper's theorem that this conclusion holds in a special case. Its general validity is presented as an open conjecture.