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More precisely, if \\(\\rho\\in[-1,0)\\), then \\(\\E[S_T^p]<\\infty\\) for every \\(0<p<p_\\rho\\), where \\(p_{-1}=\\infty\\) and \\(p_\\rho=(1-\\rho^2)^{-1}\\) for \\(-1<\\rho<0\\). For the fractional rough Bergomi kernel, this gives the finite side of the sharp critical moment threshold, complementing the known explosion result above the threshold. 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