PFA conjecture on derived models at limits of Woodin cardinals

Let $\mathcal{D}(V,\kappa)$ denote the derived model at $\kappa$, let $\Theta^{D(V,\kappa)}$ be its supremum of ordinals surjected by its reals, and let $\kappa^+$ be the successor cardinal of $\kappa$. Suppose $\kappa$ is a limit of Woodin cardinals. PFA conjecture. The following both hold:

$$\Theta^{D(V,\kappa)}<\kappa^+.$$ $$D(V,\kappa)\models AD_{\mathbb{R}}.$$

This conjecture asks how a global forcing axiom such as PFA constrains the new derived model at a limit of Woodin cardinals. It concerns both the size of the derived model's $\Theta$ and whether it satisfies determinacy for games on reals; the supplied context gives no resolution status.