Dujella's conjecture on Diophantine quadruples

Let $S=\{-4,-3,-1,3,5,8,12,20\}$. A $D(n)$-quadruple of natural numbers is a set of four distinct natural numbers such that the product of any two distinct elements increased by $n$ is a square of an integer.

Dujella's conjecture. For every $n\in S$, there does not exist a $D(n)$-quadruple of natural numbers.

For most integers outside the exceptional set $S$, existence results are known, while the cases in $S$ remain unanswered. The conjecture concerns the unresolved exceptional values of the parameter $n$.