Minimal-model conjecture under LSA

Assume $AD^+$ and that there is a largest Suslin cardinal which is a member of the Solovay sequence. A transitive model $M$ of $ZF$ is a minimal model of $\Theta strong$ when $Ord\cup\mathbb{R}\subseteq M$, $M\vDash\Theta strong$, and every transitive proper submodel $N\subsetneq M$ containing $Ord\cup\mathbb{R}$ satisfies $N\vDash\neg\Theta strong$. Here $\Theta strong$ denotes $AD_{\mathbb{R}}$ together with the assertion that $\Theta$ is a strong cardinal. Conditional minimal-model conjecture. There is a minimal model of $\Theta strong$. The hypothesis is known as LSA and is stated in the source to be consistent relative to a Woodin cardinal that is itself a limit of Woodin cardinals; the conditional existence claim remains open in the supplied text.