Miller–Tserunyan orbit-equivalence question

For every $n\in\mathbb{N}\cup\{\infty\}$ and every ergodic treeable probability-measure-preserving equivalence relation $\mathcal{R}$ of cost $n$ on a standard probability space $(X,\mu)$ , does there exist an essentially free probability-measure-preserving action $\mathbf{F}_n\curvearrowright(X,\mu)$ whose orbit equivalence relation is $\mathcal{R}$ and such that, for every $g\in\mathbf{F}_n\setminus\{e\}$ , the transformation induced by $g$ is ergodic?

Progress summary

Solved

An unrefereed August 2026 paper claims to settle the question affirmatively, but the claim has not yet been independently checked.

The question asks whether an ergodic treeable equivalence relation can be generated by an essentially free action of a free group whose nonidentity elements are all ergodic.

Known results

The edge-sliding construction gives an essentially free action of $\mathbb{F}_n$ for every treeable ergodic relation of cost $n$ , with all standard generators ergodic; the stronger all-elements question is posed there.

August 2026 claimed resolution

A new preprint claims that the required action exists, using a comeager edge-sliding argument, and presents this as settling the Miller–Tserunyan question. It is an unrefereed preprint, with no independent verification or reported objection found in the scan.

Current status (as of August 2026): The question is claimed solved by the preprint, but its proof remains unverified; no independent corroboration was found.

Sources

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Treeable equivalence-relation formulation with total weak mixing
Every ergodic treeable probability-measure-preserving equivalence relation of cost $n\in\mathbb{N}\cup\{\infty\}$ is generated by a free totally weak mixing probability-measure-preserving action of $\mathbf{F}_n$ ; that is, every nonidentity element acts weakly mixingly.
source: Wróbel, “Orbit equivalence and total weak mixing of free group actions”