The relativized mouse set conjecture for Levy-hierarchy pointclasses

Assume $ZF + AD + DC + V=L(\mathbb{R})$. Let $\alpha\in ON$ and $n\in\omega$. For each $x\in\mathbb{R}$, define $\Gamma_x$ to be the collection of all pointsets $A$ such that for some $\Sigma_n$ formula $\phi$ with parameter $x$,

$$A=\{y:J_\alpha(\mathbb{R})\models\phi[y]\}.$$

Relativized mouse set conjecture. For every $y\in\mathbb{R}$, there is an $x\in\mathbb{R}$ such that $y\leq_T x$ and $C_{\Gamma_x}$ is a mouse set. This is a boldface relativization of the preceding mouse set conjecture and is presented as an open problem connected with unreachability.