Oh ;-)

NavBar's saveState function doesn't declare any domain name for the saved cookie. This means that implicitly the domain is the domain where your page runs. You can restrict the cookie to a certain path by using the "cookie-path" preference, i.e. if you want the cookie to be available to any page inside the domain, add:

menu.prefs["cookie-path"] = "/";

before the call to “menu.generate()