The Franušić–Jadrijević conjecture on Diophantine quadruples in commutative rings

Let $\mathcal{R}$ be a commutative ring with unity and let $n\in\mathcal{R}\setminus\{0\}$. A Diophantine quadruple with property $D(n)$ in $\mathcal{R}$ is a set of four nonzero elements satisfying $a_i a_j+n=x_{ij}^2$ for all distinct indices $i,j$, with $x_{ij}\in\mathcal{R}$. Franušić–Jadrijević's conjecture. Apart from finitely many exceptions for $n$, such a quadruple exists if and only if

$$n=\alpha^2-\beta^2$$

for some $\alpha,\beta\in\mathcal{R}$. The conjecture proposes a common characterization of rings in which Diophantine quadruples exist, extending results known for several rings of integers. The paper under consideration gives infinitely many counterexamples, so the conjecture is refuted.