Has the Perfect Cuboid problem been solved?

A little over two years ago I learned of a neat unsolved problem in math. Does there exist a “perfect” cuboid – a brick-like shape with all sides and diagonals (including the long diagonal) having integer values.

The problem made for a fun Family Math project:

The problem also came up in a project inspired by a Patrick Honner post:

Mr Honner’s triangle and a surprising unsolved problem

Today Justin Lanier pointed out this post on Twitter:

I do not believe that papers on Arxiv.org have been reviewed, so for now the best I can do is hope that this paper stands up to scrutiny. It would be pretty neat to see this fun unsolved problem actually get solved.