Ok thank you. They are not breaking out the liner. Chimney is slugged full between the brick and the liner and they think it might cause problems. They are going to oval out a 6” pipe and also insulated it.
This worries me but I am not a masonry chimney expert. I like to tear them down! Perhaps a new thread about the liner installation would be helpful. There are some definite chimney guys on here. The 5x9 tiles, slugged full between brick and tile, and then trying to fit an insulated 6" liner down that seems to offer some challenges.
The fact that your clay tiles are not free floating in the masonry might explain the whole cold chimney issue!