If you’re having a conversation with a website visitor right now on live chat and they go to another page on your site (probably because you suggested it to them to help them with their problem) the chat window will be closed on the page they visit and if they open it, it opens on the home page rather than resuming the current conversation.
We’ve had a number of occasions where the visitor has been confused by this and ended up startign a new conversation - giving us a real mess in our live chat with multiple conversations open with the same person.
To make the live chat process a better experience for users it would be much better that if the visitor goes to a new page then the live chat window reopens on that page and it starts in the same conversation they were in before rather than going to the home page.
I can’t see any reason why you would want the chat window to be closed by default on the next page they visit, or start off on the home page rather than the last conversation. But making this an option in live chat settings would cover both cases.