Remember me is entirely maintained by Joomla and it's dependent on the cookie Joomla creates. The way it works is on re-visit it detects the login cookie exists. If it does it logs you back in by matching the cookie key to _user_keys and if it passes it logs you back in. If the cookie or the database row associated with the cookie is lost for whatever reason this process will fail entirely. Some browsers delete cookies on close for example or flat out reject cookies in general.
The only part CB plays in the remember me usage is sending that the checkbox was checked to Joomlas login API. For Joomla to even function properly with remember me though you must have "System - Remember Me" and "Authentication - Cookie" within Extensions > Plugins public and published.
I honestly wouldn't rely on it ever guaranteed working. About 80% of the sites I visit it doesn't function properly as the cookie gets lost at some point mostly due to browser behavior. There's hundreds of forum threads of various reports of it not working if you want to try any of the solutions proposed in those threads.
www.google.com/search?q=joomla+remember+me+not+working
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