Is it triggered after a number of comments within a short time frame, or is there a moderator button you have to click on?
I had the opportunity to answer a similar question on the Code Review meta a while back and got one of the devs to look up how we do this - the rules are pretty simple:
The link appears when two people are having a conversation with each other. So, you need exactly two people who have written at least three comments each (four on meta sites) within a four hour period. If a third (or fourth) person joins the discussion - then it won't trigger.
That doesn't mean that any additional person on the chat thread will prevent it, only in that period of six messages.
So this would trigger it (if it all happened in four hours):
Person 1, Person 2, P1, P1, P2, P2, P1
But this wouldn't:
Person 1, Person 2, P1, P1, P2, Person 3, P2, P1
Person 3 acts as a combo breaker and resets the count.
"Conversation" may be a bit specific, now that I think about it - the system doesn't care whether the two people are talking to each other - they could be both giving guidance to the OP and not interacting with each other at all but still get the encouragement to move to chat.