Please avoid extended discussions in comments. Would you like to automatically move this discussion to chat?
When I clicked the link, it—as expected—created a room, copying all my comments and the other persons's comments. However, I immediately lost access to the Teachers' Lounge. Upon further investigation, I also lost access to my chat moderation privileges.
Knowing a similar bug with reparenting was prevalent before, I checked to make sure my site association didn't change, but it was still set to Programmers.
It was only when I went back to Programmers and clicked on the chat link there did I get my moderation privileges back.
This seems like something that shouldn't occur. What happened?
With some more testing, I can do this on-demand with the link on the above mentioned answer. About 15 seconds after joining the new room, I lose access to TL.
However, it doesn't appear that merely clicking on the chat link on Programmers fixes it: I have to go to my chat profile, note that it still says Programmers under the site association, and manually select it again. Once I do that, I regain access to everything.