Equivalence of metric and unitary ergodicity

Determine whether metric ergodicity and unitary ergodicity are equivalent for measure-class preserving actions of countable groups.

Background

The paper places metric ergodicity (ME) above unitary ergodicity (UE) in the hierarchy of dynamical properties associated with weak mixing, with UE equivalent to weak mixing. Although the authors establish several implications involving chaining and metric ergodicity, they identify the equivalence of ME and UE as a notable unresolved question in the general measure-class preserving setting.