From one author's home page (Jean-Pierre Eckmann - https://fiteoweb.unige.ch/~eckmannj/publications.html), the latest version of full paper:

Walks in Rotation Spaces Return Home when Doubled and Scaled (with Tsvi Tlusty) Physical Review Letters 135, 147201 (2025)

https://fiteoweb.unige.ch/~eckmannj/ps_files/ETPRL.pdf