The conjecture on orbit equivalence relations of non-Archimedean abelian Polish groups

Let $G$ be a non-Archimedean abelian Polish group, and let a Borel action of $G$ induce an orbit equivalence relation on a Polish space. Write $E_0^\omega$ for the countable product of the eventual-equality relation $E_0$. A relation is potentially ${\bf\Pi}^0_3$ if it belongs to that Borel class after refining the Polish topology.

Orbit-relation conjecture. Every $G$-orbit equivalence relation is Borel reducible to

$$E_0^\omega,$$

and therefore is potentially ${\bf\Pi}^0_3$.

The conjecture would substantially improve the paper's general upper bound that these orbit equivalence relations are potentially ${\bf\Pi}^0_6$; the sharpness of that bound is unknown.