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Writing the coupling parameter as $ 1 / \\eps^2 $ we consider the asymptotic regime $ \\eps \\to 0 $ with the angular velocity $\\Omega$ proportional to $ (\\eps^2|\\log\\eps|)^{-1} $. We prove that if $ \\Omega = \\Omega_0 (\\eps^2|\\log\\eps|)^{-1} $ and $ \\Omega_0 > 2(3\\pi)^{-1} $ then a minimizer of the GP energy functional has no zeros in an annulus at the boundary of the disc that contains the bulk of the mass. 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