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We found evidence for strong enhancement of Cr spin fluctuations above $T_c$ in the [Cr$_{3}$As$_{3}$]$_{\\infty}$ double-walled subnano-tubes based on the nuclear spin-lattice relaxation rate $1/T_{1}$. The power law temperature dependence, $1/T_{1}T \\sim T^{-\\gamma}$ ($\\gamma \\sim 0.25$), is consistent with the Tomonaga-Luttinger liquid. 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