I think (calorie staggering and period refeeding) will help even those who are thyroid hormone dependent, and I'll explain why. Raw thyroid hormones aren't the end-all be-all of the thyroid. Even if you are taking adequate doses of synthetic thyroid hormone, if you are in a highly catabolic state (starvation), the hormones will not work properly.

As the first study I cited reported, thyroid hormones are only one half of the picture... how well they work also depends on cofators namely insulin and IGF (insulin-like growth factors). They bond to the thyroid hormones, thereby making them active and bioavailable.

WHen you trigger rapid catabolism whether by simple severe calorie reduction or a super low carb diet, insulin and IGF plummet to accomodate. This then reduces thyroid hormone potency, by resulting in less bioavailable thyroid hormone. Basically the human body has evolved over time to associate at the hip catabolism with an automatic lowering of thyroid output. Even if you're taking synthetic hormones, if you're insulin and IGF is kaput because you're practically starved, your not going to be using much energy for fuel or anabolism. In normal people with normal thyroid functioning, eating more food and sugar after prolonged deprivation will bring everything back to normal. In synthetic thyroid hormone dependent people, eating more will allow the hormones to actually work better.

That's my theory, anyway.