Steel's lightface Levy-hierarchy mouse set conjecture

Assume $ZF + AD + DC + V=L(\mathbb{R})$. A mouse set is a set $X\subseteq\mathbb{R}$ for which there is an $\omega_1+1$-iterable premouse $M$ with $X=M\cap\mathbb{R}$. Let $\Gamma$ be a level of the lightface Levy hierarchy, and let $C_\Gamma$ denote the associated canonical pointset. Steel's conjecture. The set $C_\Gamma$ is a mouse set. This asks whether every lightface Levy-hierarchy pointclass has a mouse corresponding exactly to it; the paper reports partial progress, including the case $\Gamma=\Sigma_2(J_2(\mathbb{R}))$, while the general conjecture remains open.