The hod-pair capturing conjecture

Let $obreak\textsf{AD}^+$ denote the strengthened axiom of determinacy, and let $obreak\textsf{NLE}$ denote the no-long-extender assertion. Let $obreak\textsf{HPC}$ denote hod-pair capturing: under $obreak\textsf{AD}^+$, every Suslin and co-Suslin set of reals is first-order definable over $(HC,\Sigma,\in)$ for some hod pair $(\mathcal M,\Sigma)$. HPC conjecture. Assuming $obreak\textsf{AD}^++ obreak\textsf{NLE}$, $obreak\textsf{HPC}$ holds. This is presented as a central open problem in descriptive inner model theory and, together with determinacy and $V=L({\wp}(\mathbb R))$, would imply the expected HOD analysis and GCH.