Suppose that $A \leq_a 0^\omega$ (i.e. $A$ is arithmetic in $0^\omega$) does there exist $\widehat{A} \equiv_a A$ with $\widehat{A} \leq_T 0^\omega$ [1].

More generally, say that a set $X$ is aT-complete (better name?) if every arithmetic degree $\mathbf{a} \leq_a X$ has a representative $A \leq_T X$. Does every arithmetic degree contain an aT-complete set? Do any?

[1]: In other words, if $A \leq_T 0^{\omega + n}$ does there exist $\widehat{A} \leq_T 0^\omega$, $m \in \omega$ such that $\widehat{A}^m \geq_T A$ and $A^m \geq_T \widehat{A}$