look at any connections to the thyroid gland, down and upstream.

Poor breathing = less NO, less oxygen → potential stress on thyroid metabolism (and almost any other metabolism).

NO is nitric oxide: the paranasal sinuses are a major source of NO gas.

And NO gas has antimicrobial effects (helps sterilize inhaled air), acts as a vasodilator (helps regulate blood flow and oxygen delivery), and enhances oxygen uptake in the lungs.