If P is a property of turing machines, does P $\leq_T$ COUNT_P imply that P $\in$ RE $\cup$ coRE?
Let $L \subseteq \mathbb{N}$ be a set of your choice, let $N(M)$ be the number of states in a Turing machine $M$, and let $P = \{ \langle M \rangle : N(M) \in L \}$. Then $P$ reduces to $\mathsf{COUNT}_P$, but $P$ can be arbitrarily hard.