Existence of a strong Diophantine quadruple

Problem 1.15 (Dujella, 2026). Is there any strong Diophantine quadruple, i.e. a set of four non-zero rationals $\{a_1,a_2,a_3,a_4\}$ such that $a_ia_j+1$ are perfect squares for all $1 \leqslant i,j \leqslant 4$?

Remark. There are examples of ``almost'' strong Diophantine quadruples, where all conditions for a strong quadruple are satisfied except that the condition that $a_3a_4+1$ is a square is not satisfied, e.g. $\{\frac{140}{51}, \frac{2223}{3046}, \frac{278817}{33856}, \frac{3182740}{17661}\}$ (Dujella-Petričević, Experiment. Math. 17 (2008), 83–89), but it is not known whether there are infinitely many such examples.