Do there exist positive integers $A, B, C$ such that all seven numbers $$A, B, C, A+B, B+C, A+C, A+B+C$$ are perfect squares?



3-dimensional Boolean cube of Squares
If such $(A, B, C)$ exist, then $(\sqrt{A}, \sqrt{B}, \sqrt{C})$ are sides of a perfect cuboid (see [here](https://en.m.wikipedia.org/wiki/Euler_brick)). Such a cuboid has not yet been found.


