Integer points on a curve family

Problem 4.9 (Dujella, 2026). Let $k \geqslant 3$ be an integer. Is it true that all integer points on the elliptic curve $y^2=((k-1)x+1)((k+1)x+1)(4kx+1)$ are given by $(x,y) \in \{(0,\pm 1),(16k^3-4k,\pm(128k^6-112k^4+20k^2-1))\}$?

Remark. The positive answer is known if the rank of the curve is equal to 1 (by Dujella (Acta Arith. 94 (2000), 87–101)) or $k \leqslant 5000$ (by Najman (Rocky Mountain J. Math. 40 (2010), 1979–2002)). Also if $k=3n^2+2n-2$ or $k=\frac{1}{2}(3m^2+5m)$ and the rank is equal to 2 (by Dujella (Acta Arith. 94 (2000), 87–101)).