A formula for D(-1)-quadruples

Problem 1.3 (Dujella, 2026). Let $\{a,b,c\}$ be a $D(-1)$-triple and $d$ a positive integer such that $ad+1$, $bd+1$, and $cd+1$ are perfect squares. Do all such quadruples satisfy the formula \[ (d+a+b-c)^2 = 4(ab-1)(cd+1)\ ? \]

Remark. Fujita (Acta Arith. 128 (2007), 349–375 ; Rocky Mountain J. Math. 39 (2009), 1907–1932) et He et Togbé (J. Number Theory 131 (2011), 120–137) ont obtenu la réponse positive pour les $D(-1)$-triplets de la forme $\{1,2,c\}$, $\{F_{2k+1}, F_{2k+3}, F_{2k+5}\}$ et $\{1, k^2+1, k^2+2k+2\}$, respectivement.