Szalay–Ziegler conjecture on S-Diophantine quadruples

Let $S$ be a finite set of primes. An $S$-Diophantine quadruple is a tuple of four positive, pairwise distinct integers $(a_1,a_2,a_3,a_4)$ such that

$$a_i a_j+1$$

is an $S$-unit for every $1\leq i<j\leq 4$. Let $p$ and $q$ be distinct primes, so that $S=\{p,q\}$.

Szalay–Ziegler conjecture. No $\{p,q\}$-Diophantine quadruple exists.

The conjecture is motivated by partial nonexistence results under congruence and size restrictions, together with computer verification. The source does not state whether the conjecture has been resolved.