The analytic non-orbit-equivalence conjecture for $E_1$

Let $E_1$ be the equivalence relation of eventual equality on $\b5\mathbb{N}^{\b5\mathbb{N}}$, and let analytic equivalence relations be ordered by Borel reducibility. An equivalence relation is an orbit equivalence relation if it is induced by a continuous action of a Polish group.

The analytic non-orbit-equivalence conjecture. The equivalence $E_1$ is not the least analytic equivalence relation which is not Borel reducible to an orbit equivalence.

The conjecture asks whether $E_1$ is the first obstruction to being Borel reducible to an orbit equivalence relation among analytic equivalence relations. It is known affirmatively for the special class of equivalence relations $E_\mathcal{I}$ associated with ideals, by work of Solecki, but remains open in the general analytic case.