Hjorth–Kechris dichotomy conjecture for Borel equivalence relations

Let $E$ be a Borel equivalence relation, and let $E_1$ denote eventual agreement on countable sequences of reals. For equivalence relations $E$ and $F$, write $E\leq_B F$ when $E$ is Borel reducible to $F$, and say that they are Borel bireducible when each is Borel reducible to the other. An orbit equivalence relation is the equivalence relation induced by a Borel action of a Polish group. Hjorth–Kechris conjecture. Either $E_1\leq_B E$ or $E$ is Borel bireducible with an orbit equivalence relation. This is a proposed dichotomy for Borel equivalence relations, refining the obstruction given by $E_1$. The analogous Kechris–Louveau dichotomy is known in restricted settings but is false in general; the Hjorth–Kechris version is presented as a conjecture and its status is not resolved in the supplied text.