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We prove this conjecture when the domain annulus is not too wide; explicitly, when $\\log \\frac{R}{r} \\le {3/2}$. For general $A(r,R)$ the conjecture is proved under additional assumption that either $h$ or its normal derivative have vanishing average on the inner boundary circle. This is the case for the critical Nitsche mapping which yields equality in the above inequality. 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