That's like that by design.
During a live conversation we don't want the triggers to work - trigger words for example -
It is imperative that live chat operators get some training about the importance of leaving the live chats as soon as they are finished.
That's how the statistics work too - chat operators should be encouraged to close as quickly as possible so their statistics are good. Also adding notes to the conversations so next time the same visitor comes back they can read the notes left by the previous operator.