Mouse-set conjecture for projective-like ordinals

Let $A_{(\alpha,n)}$ be the sets defined from $J_{\alpha}(\mathbb{R})$, and let a mouse set mean a set $A\subseteq\mathbb{R}$ of the form $A=\mathbb{R}\cap\mathcal{M}$ for some countable, realizable, meek premouse $\mathcal{M}$. Mouse-set conjecture. If $\alpha\geq2$ is projective-like, then $A_{(\alpha,n)}$ is a mouse set for every $n\geq0$. This is the main conjecture motivating the paper; the preceding results establish related inclusions and identifications for some values of $\alpha$ and leave the general mouse-set assertion open.