On families of cubic split Thue equations parametrised by linear recurrence sequences

Let $(A_n)_{n\in \mathbb{N}}$, $(B_n)_{n\in \mathbb{N}} \in \mathbb{Z}^{\mathbb{N}}$ be two linear-recurrent sequences that meet a dominant root condition and a few more technical requirements. We show that the split family of Thue equations $$ |X(X-A_n Y)(X-B_nY) - Y^3| = 1 $$ has but the trivial solutions $\pm\{ (0,1), (1,0), (A_n,1), (B_n,1)\}$, if the parameter $n$ is larger than some effectively computable constant.