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Theory argues that hot regions would then wander through the system and thermalize all of it.","posed_since":"2015","precise":"For the random-field Heisenberg chain at disorder $W$ where exact diagonalization at $L \\le 22$ shows an energy-dependent localization threshold $\\epsilon_c(W)$ (Luitz, Laflorencie, Alet, PRB 91, 081103, 2015), do eigenstates with energy density below $\\epsilon_c$ remain localized as $L \\to \\infty$? De Roeck, Huveneers, Muller, Schiulaz (PRB 93, 014203, 2016) argued that mobile ergodic bubbles exclude this, so MBL requires localization at all energies. Answer yes or no.","problem_ref":"3d0ebde56aa5d069ee4ba5650def899c469f5ceee1a40bbf6315c13df3a063c6","references":"","settled_by":"Time evolution of low-energy-density initial states at sizes where rare high-energy-density bubbles are typical, or a theory that closes or validates the bubble mechanism.","status_note":"The 2016 bubble argument is unrefuted while finite-size mobility edges persist in numerics; no decisive large-system result is known (moderate confidence).","title":"Can a many-body mobility edge exist in the thermodynamic 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