Mixing orbit-equivalent action

Determine whether the free totally weak mixing pmp action beta in Theorem 1 can be chosen to be mixing while preserving the orbit equivalence relation of the given free ergodic pmp action of the free group F_n.

Background

The paper proves that every free ergodic probability-measure-preserving action of a free group F_n is orbit equivalent to a free totally weak mixing action. Equivalently, every ergodic treeable pmp equivalence relation of cost n is generated by such an action.

The authors identify a natural strengthening of their theorem: replacing total weak mixing by mixing. This would require finding, within the same orbit-equivalence class, an action for which the relevant group action has the stronger mixing property. The paper explicitly records this as an unresolved question attributed to Tucker-Drob.

References

We note a natural follow-up question which remains open. \begin{question}[Tucker-Drob] \label{question:Mixing} Can $\beta$ in Theorem \ref{thm:main} be taken to be mixing? \end{question}

— Orbit equivalence and total weak mixing of free group actions  (2608.20165 - Wróbel, 20 Aug 2026) in Question (Tucker-Drob), labeled \ref{question:Mixing}, immediately following the introduction and strategy discussion

If $e \neq x \in \mathbb{F}2$, is there a free, p.m.p action $\beta$ such that $E\alpha = E_\beta$ and $x$ acts ergodically?

— Open Problems in Mathematical Logic  (2608.26628 - Barmpalias et al., 27 Aug 2026) in 2025 Section 15, item (i)

(Miller--Tserunyan). Is there a free, p.m.p action $\beta$ such that every nontrivial element of $\mathbb{F}_2$ acts ergodically?

— Open Problems in Mathematical Logic  (2608.26628 - Barmpalias et al., 27 Aug 2026) in 2025 Section 15, item (ii)