Dividing a Square into 7 Similar Rectangles

This is a continuation of my post Dividing a square into similar rectangles, in which I discussed this puzzle: if you partition a square into n similar rectangles, what proportions can these rectangles have?

Some people on Mathstodon put a lot of work into this and made some nice progress. But there’s been a surprising new twist, and I’m not even talking about the fact that the New York Times ran an article about this puzzle:

• Siobhan Roberts, The quest to find rectangles in a square, New York...