The generalized derived model theorem for regularity of Θ

Let $\delta$ be a Woodin cardinal that is a limit of Woodin cardinals. Let $g\text{?}$ be $V$-generic for $\mathrm{Col}(\omega,{<}\delta)$, and let $L(\Gamma^*_g,\mathbb{R}^*_g)$ be the derived model at $\delta$ computed in $V[g]$. Let $\mu_{\alpha}$ denote the club filter on $\wp_{\omega_1}({}^{\omega}\alpha)$. Generalized derived model conjecture. In $V(\mathbb{R}^*_g)$, both

$$L({}^{\omega}\operatorname{Ord},\Gamma^*_g,\mathbb{R}^*_g)\models\Theta\mathsf{reg}$$

and

$$L({}^{\omega}\operatorname{Ord},\Gamma^*_g,\mathbb{R}^*_g)[\langle\mu_{\alpha}\mid\alpha\in\operatorname{Ord}\rangle]\models\Theta\mathsf{reg}+\omega_1\text{ is supercompact}.$$

This would extend the known derived model results for $\mathsf{AD}^+$ and supercompactness of $\omega_1$ to stronger regularity of $\Theta$; the paper explains that the corresponding stronger determinacy assumptions were not known to be consistent from large cardinals.