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?

Are there exists such a set for a large value of $n$, say $1000$? I worked a while on this seemingly easy problem but couldn't find the answer.

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.

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 rectangular cube such that the lengths of diagonals of all of it's faces are integers.

Are there exists such a set for a large value of $n$, say $1000$? I worked a while on this seemingly easy problem but couldn't find the answer.

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?

Are there exists such a set for a large value of $n$, say $1000$? I worked a while on this seemingly easy problem but couldn't find the answer.

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 rectangular cube such that the lengths of diagonals of all of it's faces are integers.

Are there exists such a set for a large value of $n$, say $1000$? I worked a while on this seemingly easy problem but couldn't find the answer. I also saw somewhere that the similar problem is open for the case $n=3$. Is it true?

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 rectangular cube such that the lengths of diagonals of all of it's faces are integers.

Are there exists such a set for a large value of $n$, say $1000$? I worked a while on this seemingly easy problem but couldn't find the answer.