The covering conjecture for iteration sets at Suslin cardinals

Assume $AD+V=L({\mathbb{R}})$. Let $\kappa<\lambda<\Theta$, where $\kappa$ is either a Suslin cardinal or the successor of a Suslin cardinal. Let $A\subseteq{\mathbb{R}}$ be ordinal definable with $\lambda\leq\gamma_{A,\infty}$, and let $B\in{\wp}_\kappa(\lambda)$. An $A$-iteration set is a set $X\in{\wp}_\kappa(\lambda)$ such that, for every $\alpha\in X$, there is an $A$-iterable $\mathcal Q$ with $\alpha\in\operatorname{rng}(\pi_{(\mathcal Q,A),\infty})$ and

$$\pi_{(\mathcal Q,A),\infty}[\gamma_A^{\mathcal Q}]\cap\lambda\subseteq X.$$

Covering conjecture. For every such $\lambda$, $A$, and $B$, there is an $A$-iteration set $X\in{\wp}_\kappa(\lambda)$ such that $B\subseteq X$. This asserts a covering property for subsets of $\lambda$ of size less than $\kappa$ by iteration sets, under determinacy and the stated inner-model hypothesis. The source provides no evidence of resolution, so the conjecture is recorded as open.