Johnson–McClendon conjecture on continuous orbit equivalence under bounded speedups

The conjecture asks whether minimal bounded or constant speedups preserve continuous orbit equivalence for the relevant free $\mathbb{Z}^d$-odometers. The new construction claims that they do not for a class of adding-type $p$-adic systems.

Known results

• Alvin, Ash, and Ormes (2018): every minimal bounded speedup of a $\mathbb{Z}$-odometer is a conjugate odometer.
• Bezuglyi, Medynets, and others (2021): in higher dimensions, a minimal bounded speedup of a free $\mathbb{Z}^{d_1}$-odometer is a free $\mathbb{Z}^{d_2}$-odometer, but need not be conjugate to the original.
• The same 2021 work gives continuously orbit-equivalent $\mathbb{Z}^2$-odometers for which no bounded speedup in a specified cone is conjugate to the other.

September 2026 claimed counterexample

The preprint On free minimal constant speedups violating continuous orbit equivalence in $p$-adic $\mathbb{Z}^d$-odometers of adding type claims free, minimal, bounded speedups that are not continuously orbit equivalent to their originals. If correct, this disproves the conjecture under the stated mild assumptions; the claim is currently unrefereed.

Current status (as of September 2026): A counterexample is claimed for adding-type $p$-adic $\mathbb{Z}^d$-odometers, while independent verification and the full scope of the result remain open.