No, because hyperthyroidism is the situation where you have too little iodine, forcing the body to favour production of T3 over T4 because that is more Iodine cost effective. The thyroid cannot predict how much T3 the body will need that day, nor can it underproduce, because that would be life threatening, so it always overproduces T3. You're in real trouble if this slips into hypothyroidism, because then the body would not even have enough Iodine to produce enough T3.