The DLIH and BDL conjecture for generic generators

Let $obreak\textsf{AD}_{\mathbb R}$ denote determinacy for real games. Suppose that for every set $B\subseteq\mathbb R$ there is a function $F$ as in Theorem n*x such that $F$ is not pe-bad and $B\in\widetilde\Delta_{\operatorname{Base}(F)}$. In the pure extender version, assume $obreak\textsf{LEC}$ fails and set $obreak\Delta=\Delta_{\mathrm{gen},\mathrm{pe}}$; in the hod-pair version, assume $obreak\textsf{HPC}$ fails and set $obreak\Delta=\Delta_{\mathrm{gen},\mathrm{hp}}$. DLIH and BDL conjecture. In either version, both $obreak\textsf{DLIH}(\Delta)$ and $obreak\textsf{BDL}(\Delta)$ hold. The paper notes that this follows assuming $obreak\textsf{NWLW}$, while the unconditional comparison theorem needed for the conjecture remains to be proved.