Countability of id-jump traceable reals

A real is an element of $2^{\omega}$. For an order function $h$, a real $x$ is $h$-jump traceable if, for every Turing functional $\Phi$, there is an $h$-bounded computably enumerable trace $(U_n)_{n\in\mathbb{N}}$ such that, whenever $\Phi^x$ is total, $\Phi^x(n)\in U_n$ for almost every $n$. The statement concerns id-jump traceable reals, namely the case of the identity order function.

Countability conjecture. There are only countably many id-jump traceable reals.

The surrounding discussion proves instead that, for every positive rational $\epsilon$, there are $2^{\aleph_0}$ many $n^{1+\epsilon}$-jump traceable reals. The status of the asserted countability claim is not resolved in the supplied text.