The {5,q}-Diophantine quadruple conjecture

Let $q$ be a prime. The $\{5,q\}$-Diophantine quadruple conjecture. If $q\not\equiv1\pmod4$, then no $\{5,q\}$-Diophantine quadruple exists.

The authors say that this should be provable with additional effort and mention analogous possible results for $p=7$ or $p=11$; those extensions are not part of this conjecture.