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In this paper, we study the two-sample plug-in estimator of the Gaussian-smoothed Wasserstein cost \\(T_p^{(\\sigma)}(\\mu,\\nu)=W_p(\\mu*\\gamma_\\sigma,\\nu*\\gamma_\\sigma)^p\\) on \\(\\R^d\\). For fixed smoothing and finite polynomial moments \\(M_{q_\\mu}(\\mu)<\\infty\\), \\(M_{q_\\nu}(\\nu)<\\infty\\), with \\(q_\\mu,q_\\nu>p\\), we establish upper bounds in probability of order \\(\\rho_{q_\\mu,p,d}(m)+\\rho_{q_\\nu,p,d}(n)\\). 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