Model completeness conjecture for the universally Baire Chang model

Assume $\mathbf{MAX}(\mathsf{UB})$ and $(*)\text{-}\mathsf{UB}$. Let $\Theta$ be the supremum of the ordinals $\alpha$ which are the surjective image of some $\phi:2^\omega\to\operatorname{Ord}$ existing in $L(\mathsf{UB})$. Here $L_\Theta(\mathsf{UB}^{\omega_1})$ denotes the corresponding level of the constructible hierarchy with universally Baire predicates, and $\in_{\Delta_1}\cup\{\omega_1,\mathbf{NS}_{\omega_1},\mathsf{UB}\}$ is the indicated signature, where $\mathsf{UB}$ detects which subsets of $2^\omega$ are universally Baire.

Model completeness conjecture. The theory of $L_\Theta(\mathsf{UB}^{\omega_1})$ is model complete for the signature

$$\in_{\Delta_1}\cup\{\omega_1,\mathbf{NS}_{\omega_1},\mathsf{UB}\}.$$

This conjecture concerns improving the model-completeness consequences of Woodin-style axioms. It does not assert that $\Theta$ is regular in $V$; the source notes that $\Theta$ could have cofinality $\omega_2$, while conjecturally not $\omega_1$, and that an argument of Woodin combined with other results should show that $\Theta$ is not regular in models of $\mathsf{MM}^{+++}$.