That part made me raise my eyebrow, too bad not much info on it- chemically speaking, it does seem that elemental Iodine would break double bonds in unsaturated fats.
Perhaps there's a link between excess dietary fat and the displacement of Iodine from the body (particularly polyunsaturated fats since they have highest Iodine levels).
Dr Price does warn on excess polyunsaturated fats consumption (perhaps there may a link)
I wonder if the link between weight gain and hypothyroidism is due to that note (low iodine may hinder fat metabolism in the way described?).