I just found this, written by diogenes some time ago, that I saved:
The following papers from the Japanese shows that, in people with no working thyroid, "normal" healthy-type conditions on T4 therapy only occur if TSH is between 0.03-0.5.
Above this, hypo symptoms are still present. You are miles away from achieving normality.
Ito M, Miyauchi A, Hisakado M et al. Biochemical markers reflecting thyroid function in athyreotic patients on levothyroxine mono therapy. Thyroid 2017;27:484-490
Ito M, Miyauchi A, Morita S et al. TSH-suppressive doses of levothyroxine are required to achieve preoperative native serum triiodothyronine levels in patients who have undergone total thyroidectomy. Eur. J Endocrinol 2012;167:373-378
I believe someone was looking for something like this a little while ago, but I can't remember who.
Hope it helps someone.