Always log out

No, we didn't. There is a separate setting for automatically logging Facebook user's in in the JFBConnect -> Configuration area. For that to work, the user must be authenticated on Facebook.com.

None of the social networks support the 'Remember Me' functionality. It's not something that can be done; that setting only works for Joomla credentials.

From what you mentioned above:

Yes it happens with all authentication... I will try to aks directly with joomla support.

I'm not seeing that behavior. When I use Joomla credentials on your site, I am logged back in. Are you seeing something different with Joomla credentials?

I completely deactivated JFBConnect and I remain connected even when I close the browser (Chrome).

If you completely disable JFBConnect, the Facebook authentication will not work at all. How are you testing this case?

I hope that helps explain,
Alex