A Polynomial Variant of Diophantine Triples in Linear Recurrences

Let $(G_n)_{n=0}^{\infty}$ be a polynomial power sum, i.e. a simple linear recurrence sequence of complex polynomials with power sum representation $G_n = f_1\alpha_1^n + \cdots + f_k\alpha_k^n$ and polynomial characteristic roots $\alpha_1,\ldots,\alpha_k$. For a fixed polynomial $p$, we consider triples $(a,b,c)$ of pairwise distinct non-zero polynomials such that $ab+p, ac+p, bc+p$ are elements of $(G_n)_{n=0}^{\infty}$. We will prove that under a suitable dominant root condition there are only finitely many such triples if neither $f_1$ nor $f_1 \alpha_1$ is a perfect square.