Some time ago the following rather easy problem appeared in an online publication called "Problems in Elementary NT" by Hojoo Lee:

Prove that there are infinitely many positive integers $a$, $b$, $c$ that are consecutive terms of an arithmetic progression and also satisfy the condition that $ab+1$, $bc+1$, $ca+1$ are all perfect squares.

This can be done using Pell's equation. What is interesting however is that the following result for four numbers apparently holds:

Claim. There are no positive integers $a$, $b$, $c$, $d$ that are consecutive terms of an arithmetic progression and also satisfy the condition that $ab+1$, $ac+1$, $ad+1$, $bc+1$, $bd+1$, $cd+1$ are all perfect squares.

I am curious to see if there is any (decent) solution.

Thanks.