Iterates of prime producing polynomials and their Galois groups

Let $\F_q$ be a finite field of characteristic $p>0$. We prove that, given $F(t,x)\in \F_q[t][x]$ an irreducible separable monic polynomial in the variable $x$ and a generic monic polynomial $\phi(t)$ in the variable $t$, the polynomial $F(t,\phi)$ is a prime producing polynomial over large finite fields under suitable irreducible specialization. We also prove that $F(t,\phi)$ satisfies Odoni's conjecture, namely the arboreal Galois representation associated to $F(t,\phi)$ is surjective.