But it was recorded in 2008, before Circus was released. There are pictures of her in the studio with the producers of the song. And that is the only time they worked together. And for all we know it, it probably wasn't registered till around the same time as DKMW.
****. I cannot get over this song. I thought it was really silly when I first heard it, as it was nothing like I had been expecting all of these years. But the chorus has changed my mind. The entire song is flawless.