Incomparable actions of free groups

Suppose that X is a Polish space, E is a countable Borel equivalence relation on X, and μ is an E-invariant Borel probability measure on X. We consider the circumstances under which for every countable non-abelian free group Γ, there is a Borel sequence (·_r)_r ∈ℝ of free actions of Γ on X, generating subequivalence relations E_r of E with respect to which μ is ergodic, with the further property that (E_r)_r ∈ℝ is an increasing sequence of relations which are pairwise incomparable under μ-reducibility. In particular, we show that if E satisfies a natural separability condition, then this is the case as long as there exists a free Borel action of a countable non-abelian free group on X, generating a subequivalence relation of E with respect to which μ is ergodic.