Odoni's conjecture on arboreal Galois representations is false

Suppose $f\in K[x]$ is a polynomial. The absolute Galois group of $K$ acts on the preimage tree $\mathrm{T}$ of $0$ under $f$. The resulting homomorphism ${\varphi }_f:{\mathrm{G}\mathrm{a}\mathrm{l}}_K\to \mathrm{A}\mathrm{u}\mathrm{t}\mathrm{T}$ is called the arboreal Galois representation. Odoni conjectured that for all Hilbertian fields $K$ there exists a polynomial $f$ for which ${\varphi }_f$ is surjective. We show that this conjecture is false.