Hyperrectangles with integer diagonals

Question

What is the largest value of $n$ for which there exists $n$ (not necessarily distinct) complete squares of natural numbers such that the sum of every subset of it is also a complete square? ( For example, for $n=2$ this sets are two smallest elements of a Pythagorean triplet)

An equivalent form: largest $n$ for which there exists an $n$-dimensional hyperrectangle such that the lengths of diagonals of all of it's faces are integers?

Can one prove that such a set doesn't exist for a large value of $n$, say $1000$? I worked a while on this seemingly easy problem but couldn't find the answer.

Answer 1

The largest $n$ for which such a thing is known to exist is $n=2$. It is a notorious open problem as to whether such a thing exists for $n=3$. See, for example, the link given by James Cranch in the comments. If it doesn't exist for $n=3$, then it doesn't exist for any $n\ge3$. I don't think there is any $n$ for which it has been proved not to exist.