Sabok's conjecture on the complexity of isomorphism of ergodic measure-preserving transformations

Let EMPTs denote ergodic measure-preserving transformations, and consider their isomorphism relation under measure-preserving conjugacy. An equivalence relation is Borel reducible to another if there is a Borel reduction between them; a relation is a complete orbit equivalence relation if it is Borel bireducible with an orbit equivalence relation of a Polish group action. Sabok's conjecture. The isomorphism relation of EMPTs is a complete orbit equivalence relation. The conjecture concerns the exact complexity of classifying ergodic measure-preserving transformations: although their isomorphism relation is known not to be Borel, it remains open whether it is complete among orbit equivalence relations.