Solovay-sequence conjecture on club HOD-Berkeley cardinals

Assume

$${\sf{AD}}_{\mathbb{R}}+V=L(\mathcal{P}(\mathbb{R}))+“\Theta\text{ is a regular cardinal}”.$$

Let $\Theta$ be the least ordinal not surjected by a real. Suppose $\zeta<\Theta$ is a successor member of the Solovay sequence, let $\nu$ be the largest member of the Solovay sequence below $\zeta$, set

$$X=\bigcup_{\alpha<\zeta}\alpha^\omega,$$

and let

$$\Delta=\{A\subseteq\mathbb{R}:\text{the Wadge rank of }A\text{ is }<\nu\}.$$

Assume also that $\mathcal{P}(\mathbb{R})\cap{\sf{HOD}}_X=\Delta$. A cardinal $\kappa$ is a club $N$-Berkeley cardinal if for every $M\in N$ there is a club $C\subseteq\kappa$ such that for every $\alpha\in C$ there is an elementary embedding $j:M\to M$ with $\operatorname{crit}(j)=\alpha$.

Solovay-sequence conjecture. Under these assumptions, $V_\zeta^{{\sf{HOD}}_X}$ is a model of

{\sf{ZF}+“\omega_1\text{ is a club }{\sf{HOD}}\text{-Berkeley cardinal}”.

This predicts a specific inner model of determinacy and regular $\Theta$ has the stated Berkeley-cardinal property at $\omega_1$. No resolution is supplied in the source context.