Sabok's completeness conjecture for conjugacy of automorphisms

Let $(X,\mu)$ be a standard probability space. The Polish group $\operatorname{Aut}(X,\mu)$ acts on itself by conjugation, inducing an orbit equivalence relation. An orbit equivalence relation is called complete if every orbit equivalence relation is Borel reducible to it.

Sabok's conjecture. The action of $\operatorname{Aut}(X,\mu)$ on itself by conjugation is a complete orbit equivalence relation.

The claim concerns the descriptive complexity of measure-preserving automorphisms under conjugacy. The surrounding discussion notes that related conjugacy relations are complete analytic and that the conjugation action on ergodic automorphisms is turbulent, but gives no resolution of this stronger completeness claim.