Integer points on a triple-induced curve

Problem 4.8 (Dujella, 2026). Let $\{a,b,c\}$ be a Diophantine triple. Do all integer points on the elliptic curve $y^2=(ax+1)(bx+1)(cx+1)$ (such that $x \neq -1$) satisfy the condition that $ax+1$, $bx+1$ and $cx+1$ are all perfect squares?

Remark. Partial positive answers are known for the family of elliptic curves from Problem 4.9, and for families corresponding to triples of the form $\{1,3,c\}$ (Dujella and Pethő, Publ. Math. Debrecen 56 (2000), 321–335) and $\{F_{2k},F_{2k+2},F_{2k+4}\}$ (Dujella, J. Théor. Nombres Bordeaux 13 (2001), 111–124).