Uncountably many settled elements are never Frobenius elements

Boston and Jones conjectured that given a number field $K$, a quadratic polynomial $f\in K[x]$, and an element $\alpha \in K$ such that the arboreal Galois representation ${\rho }_{f,\alpha }:\text{Gal}(\bar{K}\mathrm{/}K)\to \text{Aut}(T_{\mathrm{\infty }})$ attached to the pair $(f,\alpha )$ is well defined, any element in the image of such representation that is a Frobenius for an unramified prime of $K$ is a settled element. We exhibit an explicit uncountable subset $S\subseteq \text{Aut}(T_{\mathrm{\infty }})$, consisting of pairwise non-conjugate settled elements, such that no matter how $K,f,\alpha$ are chosen, no element of $S$ can be a Frobenius for any ramified or unramified prime of $K$.