Wilson's conjecture on the first Solovay ordinal of a derived model

Let $V$ satisfy PFA, let $\nobreak\bkappa$ be a limit of Woodin cardinals, and let $D(V,\kappa)$ denote the derived model at $\kappa$. Write $\Theta_0^{D(V,\kappa)}$ for the first level of its Solovay hierarchy. Wilson's conjecture. Assume PFA and that $\kappa$ is a limit of Woodin cardinals. Then

$$\Theta_0^{D(V,\kappa)} < \kappa^+$$

and

$$\Theta_0^{D(V,\kappa)} < \Theta^{D(V,\kappa)}.$$

Wilson's original conjecture removes the countable-cofinality assumption from his earlier theorem bounding the first Solovay ordinal. The paper presents this as an open generalization; the supplied source gives no resolution.