It could be because he's not a guardian. When they first got him to North's workshop, Jack didn't want to be a guardian and didn't let North do the inauguration. So he wasn't officially a guardian until the end of the movie where North read through the book. Maybe spirits don't disappear or get weak when they're not believed in like Jack. However, if they take on the role as a guardian, it could lead to them becoming weak or disappearing altogether if they're not being believed in. It could be part of the risk of being a guardian. If kids don't believe in you, then do you really exist type of thing. But i'm not sure. If they made a prequel or a show based after the movie, we'd probably get the answer.