Conjecture on the height of the derived cardinal

Let $(\mathcal{V},\Omega)$ be an excellent least branch hod pair with $\mathcal{V}\models\mathsf{ZFC}$, let $\delta$ be a Woodin limit of Woodin cardinals in $\mathcal{V}$, and let $\mathcal{Q}$ and $\eta$ be as in the stated conclusion of the main result. Write $\delta^{\mathcal{Q},\eta}_{\infty}$ for the associated derived cardinal. Conjecture on $\delta^{\mathcal{Q},\eta}_{\infty}$. The cardinal $\delta^{\mathcal{Q},\eta}_{\infty}$ is weakly inaccessible and greater than $\Theta$ in $\mathsf{CDM}^+(\mathcal{Q},\eta)$. The main theorem establishes that this cardinal is at least $\Theta$; under regularity assumptions on $\delta$, regularity is known, while weak inaccessibility strictly above $\Theta$ remains conjectural.