Wouldn't this make it obvious that the user is browsing in private mode? While I get that might be preferable to leaking information, its also not an ideal solution either.
I'm very interested in this thread. Other replies here are correct - there are many ways that sites try to detect private browsing, and many ways they can get it right or wrong.
How do people feel about the "stealth" design goal of private browsing? Should it be a goal? What about a hide-in-a-big-crowd tactic? (E.g., how Tor tries to make all its users look identical.)