Feel you, i think there should maybe be a button like on discord like "Busy, Online, Offline" but in the terms of the PM's
For example "Busy Status: This person is currently busy, try again later!"
"Offline Status: This player is offline (as it says already)
"Online Status" This player is currently online and might answer soon!
But whenever leaving an Online Status chat, it will say "This player left the PM" so they know the conversation is broken
But when u in an active chat and switch to "Busy Status" it will send a message "This player went to go Busy" and when u leave the chat than it wont show orsomething?