Odoni's conjecture on surjective arboreal representations

Let $K$ be a Hilbertian field of characteristic $0$, and let $n\geq 2$. For a polynomial $f\in K[x]$, write $f^{-k}(0)$ for the set of $k$-th preimages of $0$, and let $[S_n]^k$ denote the $k$-fold wreath product of $S_n$ with its natural action on the rooted $n$-ary tree of height $k$. Odoni's conjecture. There exists a polynomial $f(x)\in K[x]$ of degree $n$ such that

$$\operatorname{Gal}\left(K\left(f^{-k}(0)\right)/K\right)\simeq [S_n]^k$$

for every $k\geq 1$; equivalently, the associated arboreal representation $\rho_{f,\infty}$ is surjective. The conjecture is false for general Hilbertian fields of characteristic $0$, although it motivates positive results for suitable fields.