Well, I've tried it in multiple browsers, all show the same problem, also on CB 2.0. This is J2.5.27
The site is running https, I've selected the 'Remember Me' and when I start the browser it shows as not logged in, and if I refresh with f5, it shows the right state.
Chrome:
screencast.com/t/QZSHYLY3p
Internet Explorer:
screencast.com/t/wmGJa3yDQd
Firefox:
screencast.com/t/xzthz6KylO
I don't have any cookie control, the only 'cookie widget' I have installed is the EUCookieDirective plugin, and disabling that doesn't change anything.