How come? I never use private mode and assumed it simply clears cookies and history after the session ends. Is there more to it?
Try clicking this link and see what it says :) https://www.whatismybrowser.com/detect/what-is-my-referrer/
Keyword being "a little". It's a cookie set on previous visit.
$ document.cookie
> HN=1
Delete the cookie, then open new tab, copy-paste the link and it works just fine. It's not rocket science, at least it didn't used to be :)