The Chang-model cardinal-regularity conjecture

Let $V$ be a hod mouse and let $\lambda$ be a Woodin limit of Woodin cardinals. Let $g\subseteq\operatorname{Coll}(\omega,<\lambda)$ be $V$-generic, and let $(j,M)$ be as in the cited Chang-model theorem. Set $\nu=j(\lambda)$ and $N=j(V_\lambda)$.

Cardinal-regularity conjecture. Then

$$L_\nu(N^\omega,\Gamma_g,\mathbb{R}_g)\vDash\mathsf{ZF},$$

and

$$L_\nu(N^\omega,\Gamma_g,\mathbb{R}_g)\vDash “\text{for every cardinal }\eta\geq\Theta,\\ \eta^+\text{ is a regular cardinal}”.$$

The conjecture concerns the structure above $\Theta$ in a determinacy model arising from a hod mouse. The paper presents this as an important question; its general validity remains open.