The L[vec E]-capturing conjecture

Let $obreak\textsf{AD}^+$ denote the strengthened axiom of determinacy, and let $obreak\textsf{NLE}$ be the no-long-extender assertion. Let $obreak\textsf{LEC}$ denote $L[\vec E]$-capturing: for every Suslin and co-Suslin set of reals, there is a pure extender pair $(\mathcal M,\Sigma)$ such that the set is first-order definable over $(HC,\Sigma,\in)$. LEC conjecture. Assuming $obreak\textsf{AD}^++ obreak\textsf{NLE}$, $obreak\textsf{LEC}$ holds. The \text identifies this as a major open problem in descriptive inner model theory and states that $obreak\textsf{LEC}$ implies the corresponding hod-pair capturing principle.