This paper seems to be progress: https://link.springer.com/article/10.1007/s10898-018-0711-5

All I've concluded so far is that it must be harder than it looks to prove it. I don't think there's much doubt that it's true though.

The way you describe the problem, surely the optimal solution is to pack them on top of each other

@JussiVilleHeiskanen Seems to me it's pretty much the same terminology as you use in /JussiVilleHeiskanen/is-the-best-packing-of-squares-insi

If I "pack" a bunch of suitcases into the back of a car, then the space taken up by different suitcases can't intersect.

@JussiVilleHeiskanen Agree

The site says only n=1 and n=2 are proved optimal, so in theory this is an "open problem". But unlike most open math problems posted here, it seems simple enough that someone in the comments could produce a proof. Manifold, I'm counting on you!