What was the relationship between the duke and fake blonde?

Isnt he the one "taking care of the refound princess" tho for a stranger she only meet a few months ago.. she is trangly close to him..

Got a feeling she was "missing" while secretly hiding at the DUke's place..
After all what better ruse to gain sympathy than to be the only gifted child of the emperor who "vanished" and then returned with all the skills of a trained nobel daughter.. to one up her powerless sister..