Arithmetic ergodicity dichotomy for countable Borel equivalence relations

Let $E$ be a countable Borel equivalence relation, and let $\equiv_A$ denote arithmetic equivalence equipped with the arithmetic cone measure. Arithmetic ergodicity conjecture. Exactly one of the following holds: $E$ is weakly universal, or $\equiv_A$ is $E$-ergodic with respect to the arithmetic cone measure. This is presented as a weaker analogue of the consequence of Martin's conjecture for Turing equivalence. The conjecture is open.