Two-prime S-Diophantine quadruple conjecture

Let $p<q$ be primes and let $S=\{p,q\}$. An $S$-Diophantine quadruple is a quadruple of distinct positive integers such that every pairwise product plus one has all prime divisors in $S$. Two-prime $S$-Diophantine quadruple conjecture. No $S$-Diophantine quadruple exists.

This is equivalent to the paper's conjecture that $s(2)=4$, where $s(k)$ is the smallest integer $m$ such that no set of $k$ primes admits an $S$-Diophantine $m$-tuple. The paper presents this as an unresolved question.