When I first saw the pics I thought oil substance and soil. if a grease removing dish soap does nothing and it stays then I have no clue what would help specially if it's greasy. But I wonder if it's a oil gland issue or maybe a feather/skin disorder. due to poor breeding/ conditions of the parents. the closest I could find was wet feather in ducks but if it was you might know whats up already. but the seller sounds like a jerk and I would not work with them again specially if they are being so defensive about you asking for their input.