  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 \[
  |X(X-A_n Y)(X-B_n Y) - Y^3| = 1 \] 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.
