A subproblem of Problem D18 in the 2004 softcover reprint of the 3rd edition of Guy's *Unsolved Problems in Number Theory* asks if there are four squares the absolute values of whose differences in pairs are squares (see p. 278 in the book, the subproblem is titled "Four squares whose differences are square"). Guy comments that "Although a solution is unlikely, there do not appear to be any congruence conditions which forbid it." I have two (main) questions:

> **(Q1)** What is the status of the problem? Has there been any substantial
> progress since 2004?
> 
> **(Q2)** What about the existence of an integer $N \ge 4$ such that
> there is no $N$-tuple $(a_1, \ldots, a_N)$ of positive
> integers with the property that $|a_i^2 - a_j^2|$ is a square for all $i, j = 1, \ldots, N$? More precisely, what about $N = 5$?

The question is motivated by a problem in graph theory posed by Sergey Goryainov. I'm mainly looking for references.