There is something about iodine and TMJ. It's the only thing that stops my afternoon jaw lock! I am also hypothyroid and take Natural Dessicated Thyroid in the mornings (which contains iodine I think) and that lasts until the late afternoon. Then when those meds wear off, jaw sticks and locks I use iodine drops (potassium Iodine - lugol's) about 4 drops. Problem gone till next day. Very effective treatment but not a cure.
You are correct about iodine treating temporomandibular joint disorder. My wife had TMJ severely at times and we started taking iodine years ago and have recently increased our dosage. We have been experiencing benefits we could never imagine -all positive of course. And one of them is her TMJ is almost completely resolved. What's crazy is that the dental establishment does not even recognize iodine as a treatment.
My wife has had bad TMJ for years, leading to awful headaches as well. A $30 ebook bought online provided exercises which helped a lot - but not a permanent cure. Lately she has been taking iodine to help her low thyroid issues. She has discovered that when she takes 3 drops of iodine a day (2 drops of Lugol's Iodine and 1 drop of Atomidine) 1 drop at a time several hours apart, she has no jaw problem. Maybe TMJ can be a sign of iodine deficiency in at least some cases? Oh yes, we also quit using fluoride toothpaste decades ago, and quit drinking fluoridated water recently. Fluorine, chlorine (both in water) and bromine (in white flour) all interfere with iodine absorption.
Remember, your mileage may vary! If you take iodine and get tension, palpitations, runny nose, frontal headache - back off, you are taking too much for your present needs!