This is incredible!!! My first paper in my PhD back in 2018 was on this problem (getting a 2^{logstar n} dim bound); since then, I have been thinking about it on and off for at least 7 years.
It's really crazy how things turned out. Last month at OpenAI, our model disproved the famous unit distance conjecture, and the number-theoretic techniques are EXACTLY what you need to settle this furthest pair problem!!!
I am so excited about the future of TCS and can't wait to see more open problems I care about being resolved by collaboration between AI and humans :)