Three-dimensional hanging curve conjecture

A thin stick with a thickness of 0 and unit length turns once in 3D space; the volume it sweeps over can be minimized to 0, and the shape formed by its swept area has both Minkowski dimension and Hausdorff dimension equal to 3.

It’s a very concise question, but after analyzing it, I realized that this seemingly simple question actually requires such a large amount of thinking.

I didn’t understand the solution to the three-dimensional conjecture—it requires mathematical tools I don’t have—but I had ChatGPT carry me through the proof of the two-dimensional hanging curve conjecture, and it was interesting.