Normally the URL at the top of the browser will have the field of play/platform (fop) explicitly listed
http://192.168.1.100:8080/displays/attemptBoard?
fop=B and that is what matters.
If on a given laptop you open the displays page, switch to the platform you're on at the top of the page, and then start the screen, it should all work.
Changing the letter on the URL live is also supposed to work.
If this happens again, write down the time and send the logs to
owlcm...@jflamy.dev (to get at the logs, use the Files menu for the version you are running).