It wouldn't be the desktop, would it? Wouldn't it be an 'empty' browser window? Still just as suspicious, of course, but I wonder if some/all browsers do something special in that case—e.g. default to the home page. They certainly could, as could a plugin.
Chrome closes the window on the last tab. It's splitting hairs, however. As you said, it's still raises suspicion which, to a person in a domestic violence situation, is not what they want.
