Unbalancedness of the conjugacy relation of ergodic measure-preserving transformations

We show that the isomorphism of ergodic measure-preserving transformations is not Borel reducible to the relation induced by the conjugacy action of the full group of an ergodic measure-preserving transformation on itself. This answers a question of Le Maître in the negative and gives a positive indication towards a conjecture of Sabok. In fact, we prove that the isomorphism of ergodic measure-preserving transformation is unbalanced, which answers another question of Le Maître in the positive.