The Solovay-sequence Nairian-model conjecture

Assume $\mathsf{AD}^+$. Let $(\theta_\alpha:\alpha\leq\Omega)$ be the Solovay sequence. Suppose that $\alpha+1\leq\Omega$ and that $\operatorname{HOD}\vDash “\theta_{\alpha+1}$ is a limit of Woodin cardinals”. Set

$$\delta=\theta_{\alpha+1},\qquad M=V_\delta^{\operatorname{HOD}},\qquad N=L_\delta\left(\bigcup_{\xi<\delta}(M|\xi)^\omega\right).$$

Solovay-sequence conjecture. Then $N\vDash\mathsf{ZF}$, $\Theta^N=\theta_\alpha$, $N$ satisfies that every cardinal $\eta\geq\Theta$ has regular successor $\eta^+$, and, for every $\eta<\delta$,

$$\operatorname{cf}((\eta^+)^N)\geq\theta_\alpha.$$

The conjecture gives a purely $\mathsf{AD}^+$ formulation of the cardinal-regularity prediction. The paper verifies a version in a restricted setting, while the full generality remains open; conclusion (1) had also been claimed independently in the HOD-analysis context.