When I close the lid of my M2 macbook air (macos Monterey Version 12.6.1), my mouse MX Anywhere 3 is removed from the Logi Options+ main menu which result of not being able to use any custom functionality of the mouse.
The only way of having the mouse back into the menu is to re-install Options+. Unfortunately, as soon as I close the lid, or after few time, the mouse is disappearing again.