Boston–Jones conjecture on settled elements in quadratic arboreal representations

Let $f(x)\in\mathbb{Q}[x]$ be a quadratic polynomial, let $\alpha\in\mathbb{Q}$, let $T$ be the binary tree, and let

$$\rho_{f,\alpha}:\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\operatorname{Aut}(T)$$

be the associated arboreal representation. An element of its image is called settled when it has the settled-element property for the action on $T$. Boston–Jones conjecture. The set of settled elements is dense in the image of $\rho_{f,\alpha}$. This conjecture concerns the abundance of elements with controlled behaviour on the levels of the binary tree and is part of the study of arboreal Galois representations. The supplied text gives no resolution, so the conjecture remains open.