LSA conjecture for a strong-cardinal model with a Woodin cardinal

Let $LSA$ abbreviate the assertion that $AD^+$ holds and there is a largest Suslin cardinal belonging to the Solovay sequence. Let $\vec{E}$ be an extender sequence and write $L^{\vec{E}}(\mathcal{P}(\mathbb{R}))$ for the corresponding extender model over the power set of the reals. LSA model conjecture. Assume $LSA$. Then there is a model $\mathcal{M}=L^{\vec{E}}(\mathcal{P}(\mathbb{R}))$ of $AD_{\mathbb{R}}$ in which $\Theta$ is a strong cardinal and $\mathcal{M}\vDash$ “there is a Woodin cardinal”. The source presents this as a conjectural direction for the fine structure of models of LSA; no proof or resolution is given.