The inclusion of the second-order constructible universe in the first-order constructible universe

Let $C(\mathop{\text{aa}})$ and $C^2(\omega)$ denote the inner models defined in the paper, with $C(\mathop{\text{aa}})$ the model generated using the relevant class of structures and $C^2(\omega)$ the second-order constructible universe. Inclusion conjecture. Assuming appropriate large cardinals, we have

$$C(\mathop{\text{aa}})\subset C^2(\omega).$$

This is a proposed relationship between the two inner models. The surrounding text gives related positive results under a proper class of Woodin cardinals and under $\text{MM}^{++}$ for reals and subsets of $\omega_1$, but does not resolve the full inclusion.