There is no communication of the sort you describe between these products whether they are of the same brand or not.
If you are connected to the the headphone output of the TV then disconnecting that cable to the transmitter would turn the TV speakers on in most cases.
I actually have to go into my TV settings to turn off the TV speaker SK I get
Output from the headphones.
So in other words if I don't turn off the speaker, no headphone sound
Different brand BTW
Do I need to turn off the speaker?