Opening greeting
When to use: Use it as your very first reply the moment a fresh chat lands in your queue.
Script
Hi visitor first name, I am agent name over at business name -- a real person here, not a bot.
I can see you are on the page topic page. What are you trying to sort out? I can give you a straight answer in a minute or two.
If they go quiet
No rush at all -- I will keep this window open. If it is easier, just tell me the outcome you want and I will do the digging for you.
Why it works
It names a human, references the exact page they landed on, and asks one open question aimed at their goal instead of a hollow offer to assist. That combination lowers the guard most visitors bring to a chat bubble and gets them talking about what they actually need.