Autostability of hyperarithmetic models

Let $\mathscr M$ be a $\Delta^1_1$-constructivizable model. If its Scott rank $\mathrm{sr}({\mathscr M})$ is strictly less than $\omega_1^\mathrm{CK}$, then it can be proved that it is autostable. If $\mathrm{sr}({\mathscr M})=\omega_1^\mathrm{CK}$, then there exists an ordinal $\alpha<\omega_1^\mathrm{CK}$ such that for all $\gamma>\alpha$, $\mathscr M$ is not autostable in any degree $0^{(\gamma+1)}$. In addition, we consider problems of the $\Delta^1_1$-autostability of $\Delta_1^1$-constructivizable Boolean algebras.