Decomposable polynomials in second order linear recurrence sequences

We study elements of second order linear recurrence sequences $(G_n)_{n= 0}^{\infty}$ofpolynomialsin$$\mathbb{C}$[x]$ which are decomposable, i.e. representable as $G_n=g\circ h$ for some $g,h\in \mathbb{C}[x]$ satisfying $\mathrm{deg}g,\mathrm{deg}h>1$. Under certain assumptions, and provided that $h$ is not of particular type, we show that $\mathrm{deg}g$ may be bounded by a constant independent of $n$, depending only on the sequence.