The Search for The Perfect Box

The Mystery
    The Perfect Box as its known, is a box whose edges are all integers, whose face diagonals are integers and it's body diagonal is an integer also.  No one has yet found a Perfect Box, but no one has yet proven that it can't exist.

The History
    The question of the box's existence has been around since about 1719.  The smallest box with all integer face diagonals, but not an integer body diagonal was discovered around then, it has measurements 44 x 117 x 240.  In all that time no one has come up with a proof as to why it can't exist.  But some facts are known about it, should it exist: