I don't know if you checked out that other thread but the bank will know (or may have suspicion) once you cease TRS on the old place, or commence TRS on a new place. Plus there will (should) be a change of insurance, which the bank will be notified of as well (banks are usually named as insured on policies and, if the policy changes, they will be notified).
It's pot luck. Some (as mentioned on that other thread) did, in fact, notify the bank and nothing happened with their interest rate. Some (like Steve D's LL) did not notify and got charged back-interest.
You should check your mortgage terms and then weigh up the risk.
I dont have TRS left.