Cold sore healed but left a red mark. Still contagious?

My partner had a cold sore over a month ago. It was pretty big and swollen for a while. It healed but it left this red mark on their lip that’s pretty noticeable and hasn’t gone away. It’s been there for a while, I tried putting Neosporin on it and waiting but it doesn’t seem to go away. Are they still contagious with this and how to get rid of it?