What single incident proves the king and the duke to be nearly as stupid as the towsnpeople? -chapter 30

I think the it would be when the King completely makes up a tattoo that would be on the diseased man's body. Clearly by now the King and Duke should cut their losses and run but they continue the charade. THe King claims the tattoo on the dead man's chest is a thin, blue arrow. He of course has no way of knowing.