Quantum diffusion for all times in the weakly disordered 3D lattice

For $H = \Delta + \lambda V$ on $Z^3$, $\Delta$ the nearest-neighbour hopping operator, show that $E \sum_{x} \mid x \mid^{2} \mid \langle x \mid \mathrm{exp}(-itH) \mid 0\rangle \mid^{2} \sim D(\lambda) t$ as $t \to \infty$ with a diffusion constant $D(\lambda) > 0$ for small $\lambda$ (or the analogous statement for an energy-localized initial state). Erdos, Salmhofer and Yau proved diffusion up to times $t \sim \lambda^{-2-\kappa}$ for some $\kappa > 0$. Answer: a proof for $t \to \infty$ at fixed small $\lambda$.

A proof of diffusive spreading uniformly in time at fixed small $\lambda$.