Wieferich-free two-prime S-Diophantine quadruple conjecture

Let $p<q$ be primes. A pair $(p,q)$ is called a Wieferich pair when it satisfies the paper's Wieferich condition. Wieferich-free two-prime $S$-Diophantine quadruple conjecture. If $(p,q)$ is not a Wieferich pair and it is not the case that $p\equiv q\equiv 1\pmod 4$, then no $\{p,q\}$-Diophantine quadruple exists.

The authors state that the case $q\equiv3\pmod4$ appears approachable and hope to prove this conjecture in forthcoming work; the Wieferich condition remains an obstacle.