Generic quadratic arboreal surjectivity conjecture over the rational function field

Let $K=\mathbb{F}_p(t)$), let $T$ be the complete infinite rooted binary tree, and let $f_t=x^2+t\in K[x]$. Write $G_{f_t}(0)$ for the arboreal Galois group associated with the orbit of $0$. Generic quadratic arboreal surjectivity conjecture.

$$G_{f_t}(0)=\operatorname{Aut}(T).$$

The preceding theorem establishes the relevant density-zero result, while this stronger assertion is proposed as likely; its resolution is not supplied in the source.