Jones's surjectivity conjecture for the arboreal representation of $x^2+t$

Let $k$ be a field of characteristic different from $2$, and let $t$ be transcendental over $k$. For the polynomial

$$\phi=x^2+t\in k(t)[x],$$

let its arboreal representation be the natural Galois action on the rooted tree of iterated preimages of a base point. Jones's surjectivity conjecture. The arboreal representation of $\phi$ is surjective. This conjecture predicts maximal Galois action for the simplest non-isotrivial quadratic polynomial over a rational function field; the source presents it as the conjecture to be proved using its equivariant reduction theorem, but the supplied text gives no resolution status.