Greatest common divisors of iterates of polynomials

Following work of Bugeaud, Corvaja, and Zannier for integers, Ailon and Rudnick prove that for any multiplicatively independent polynomials, $a, b \in {\mathbb C}[x]$,thereisapolynomial$h$suchthatforall$n$, we have $$\gcd (a^n-1,b^n-1)\mid h$$ We prove a compositional analog of this theorem, namely that if $f,g\in \mathbb{C}[x]$ are nonconstant compositionally independent polynomials and $c(x)\in \mathbb{C}[x]$, then there are at most finitely many $\lambda$ with the property that there is an $n$ such that $(x-\lambda )$ divides $\gcd(f^{\circ n}(x) - c(x), g^{\circ n}(x) - c(x))$.