The generic-generator non-badness conjecture

Let $obreak\textsf{AD}^+$ denote the strengthened axiom of determinacy. For a function $F$ as in the paper's Theorem nx, call $F$ pe-bad when, for a cone of reals $x$, the supremum of the Woodin cardinals of the associated $L[\vec E]$ construction equals $obreak\delta_x$; define hp-bad analogously. Call $F$ projective-like when its base pointclass has the stated projective representation. Generic-generator non-badness conjecture. Assuming $obreak\textsf{AD}^+$, if $F$ is as in Theorem nx and is projective-like, then $F$ is neither pe-bad nor hp-bad. The \text says that hod-pair capturing \implies this conjecture for many projective $F$, but does not establish it in full.