Looks like that when using https, the Referer header is not being sent to the server, thus http://chat.stackexchange.com/https://chat.stackexchange.com/ can't know where you came from, and won't show the proper rooms.
If you're browsing the non-secure site e.g. https://security.stackexchange.com/ it's working fine.
This is by design behavior of browsers, according to this: Is HTTP header Referer sent when going to a http page from a https page?
So, we need to ask the team to change the code so that a querystring parameter will be sent, and not rely on the Referer anymore, if we want this to work properly in https.