Wieferich-free two-prime S-Diophantine quadruple conjecture without congruence restrictions

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, then no $\{p,q\}$-Diophantine quadruple exists.

The paper describes this as a weaker form of its preceding conjecture because it removes the congruence restriction $p\equiv q\equiv1\pmod4$, and presents it as an open problem for future work.