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The question is whether, for large enough W, an infinitely long chain stays out of equilibrium forever.","posed_since":"2006","precise":"$H = \\sum_i J S_i . S_{i+1} + h_i S^{z}_i$ with $h_i$ independent and uniform in $[-W, W]$. Does a $W_c < \\infty$ exist such that for $W > W_c$ infinite-temperature eigenstates obey an area law and local memory of the initial state persists as $t \\to \\infty$, in the limit $L \\to \\infty$? Answer yes or no, with an estimate or bound on $W_c$.","problem_ref":"0cf86e50dcb1aedcccd0a8b2603b2d6cd8ba2dc41887681e77ef1284e7e962fe","references":"","settled_by":"A proof for this model (Imbrie's 2016 proof covers a random Ising chain with transverse field and relies on an unproven limited-level-attraction assumption), or numerics that control rare thermal regions at sizes where avalanche effects are visible.","status_note":"Avalanche studies (Morningstar et al. 2022, arXiv:2107.05642) place any transition at disorder well above the exact-diagonalization crossover W of about 4J; existence itself remains debated in 2025-2026.","title":"Is the random-field Heisenberg chain localized at strong 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