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A proof exists only near the edges of the energy band or at strong disorder.","posed_since":"1979","precise":"For $H = \\Delta + \\lambda V$ on $\\ell^2(Z^2)$, $\\Delta$ the nearest-neighbour hopping operator (spectrum $[-4, 4]$ at $\\lambda = 0$), $V$ i.i.d. with bounded density (or Bernoulli), prove that for every $\\lambda > 0$ the spectrum is almost surely pure point with exponentially decaying eigenfunctions at all energies, with localization length expected to grow as $\\exp(c/\\lambda^2)$. 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