Finite index theorems for iterated Galois groups of unicritical polynomials

Let $K$ be the function field of a smooth, irreducible curve defined over $overline{mathbb{Q}}$. Let $fin K[x]$ be of the form $f(x)=x^q+c$ where $q = p^{r}, r ge 1,$ is a power of the prime number $p$, and let $betain overline{K}$. For all $ninmathbb{N}cup{infty}$, the Galois groups $G_n(beta)=mathop{rm{Gal}}(K(f^{-n}(beta))/K(beta))$ embed into $[C_q]^n$, the $n$-fold wreath product of the cyclic group $C_q$. We show that if $f$ is not isotrivial, then $[[C_q]^infty:G_infty(beta)]<infty$ unless $beta$ is postcritical or periodic. We are also able to prove that if $f_1(x)=x^q+c_1$ and $f_2(x)=x^q+c_2$ are two such distinct polynomials, then the fields $bigcup_{n=1}^infty K(f_1^{-n}(beta))$ and $bigcup_{n=1}^infty K(f_2^{-n}(beta))$ are disjoint over a finite extension of $K$.