We actually know more. Autoimmune thyroid, whether hashi or Graves are both related to celiac disease or non-celiac gluten sensitivity. https://www.ncbi.nlm.nih.gov/pubmed/27784959
We also know that certain pathogens in the gut a can trigger thyroid issues. This may be connectedto intestinal permeability https://www.ncbi.nlm.nih.gov/pubmed/27307072
I also know that soy is toxic to thyroid hormones and anyone with autoimmune thyroid should avoid it.
Through Apex seminars I have data that most grains are "molecular mimics" to gluten which is a problem in celiac or non-celiac gluten sensitivity. And therefore grains should be avoided. Other molecular mimics for thyroid are spinach and tuna of all things
Chronic stress affects the HPA axis (hypothalamus pituitary adrenal axis) which can then affect the HPT axis (hypothalamus pituitary thyroid axis) according to classic physiology. Therefore chronic stress is detrimental to thyroid hormones. Chronic stress also impacts the cellular junctions of the intestinal lining which can then increase intestinal permeability leading to increase of celiac disease or non-celiac gluten sensitivity increasing autoimmune thyroid problems.
You can look up paleo autoimmune protocol for many foods to avoid that can also have an impact in intestinal permeability and autoimmunity reactions.
I hope that this list of potential factors helps. I am always happy to answer questions about this as autoimmune diseases are a focus of my practice