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We considerably improve upon the recent result of Martinelli and Toninelli on\nthe mixing time of Glauber dynamics for the 2D Ising model in a box of side $L$\nat low temperature and with random boundary conditions whose distribution $P$\nstochastically dominates the extremal plus phase. An important special case is\nwhen $P$ is concentrated on the homogeneous all-plus configuration, where the\nmixing time $T_{mix}$ is conjectured to be polynomial in $L$. In [MT] it was\nshown that for a large enough inverse-temperature $\\beta$ and any $\\epsilon >0$\nthere exists $c=c(\\beta,\\epsilon)$ such that $\\lim_{L\\to\\infty}P(T_{mix}\\geq\n\\exp({c L^\\epsilon}))=0$. In particular, for the all-plus boundary conditions\nand $\\beta$ large enough $T_{mix} \\leq \\exp({c L^\\epsilon})$.\n Here we show that the same conclusions hold for all $\\beta$ larger than the\ncritical value $\\beta_c$ and with $\\exp({c L^\\epsilon})$ replaced by $L^{c \\log\nL}$ (i.e. quasi-polynomial mixing). The key point is a modification of the\ninductive scheme of [MT] together with refined equilibrium estimates that hold\nup to criticality, obtained via duality and random-line representation tools\nfor the Ising model. In particular, we establish new precise bounds on the law\nof Peierls contours which quantitatively sharpen the Brownian bridge picture\nestablished e.g. in [Greenberg-Ioffe (2005)],[Higuchi (1979)],[Hryniv (1998)].\n