Seven squares from three numbers

Problem 5.17 (Dujella, 2026). Are there infinitely many triples of distinct non-zero rationals $(a,b,c)$ such that $a+1$, $b+1$, $c+1$, $ab+1$, $ac+1$, $bc+1$, and $abc+1$ are all squares?

Remark. Some triples with the given property are $(8, \frac{15}{49}, \frac{1320}{49})$, $(8, \frac{312}{529}, \frac{495}{529})$, $(\frac{25}{144}, -\frac{143}{144}, \frac{1792}{3249})$. By the recent paper Dujella, A., Szalay, L.: Four squares from three numbers. Preprint (2025). arXiv: 2506.14013, there are infinitely many triples $(a,b,c)$ of integers greater than 1 such that $ab+1$, $ac+1$, $bc+1$, and $abc+1$ are all squares.

Problem solved in the recent paper Dujella, A., Kazalicki, M., Petričević, V.: Seven squares from three numbers. Preprint (2026). arXiv:2604.08729. The solution is based on the fact that the quartic $3u^4-9u^2+7=z^2$ has infinitely many rational points. This gives an infinite sequence of triples with the given property, starting with $(8,\frac{312}{529},\frac{495}{529})$, $(\frac{312}{529},-\frac{152880}{165649},-\frac{78374557}{87628321})$, $(-\frac{152880}{165649},\frac{724255280}{736742449},-\frac{63009087694401}{122040649934401})$, $(\frac{724255280}{736742449},\frac{24490915482072}{12448992625969},-\frac{4510665894525110607837}{9171701314839342058081})$, $(\frac{24490915482072}{12448992625969},-\frac{2539564321528123032}{5054545907282329441},\frac{14261842404349331345950974819695}{62924004727379507987985949853329})$. In the same paper, it is proved that there does not exist a triple of non-zero integers with the same property.