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Phys. \\textbf{5}, 321 (2009)] found a surprisingly large, 2-3 orders of magnitude larger than usual $^{13}$C hyperfine coupling (HFC) in $^{13}$C enriched single-wall carbon nanotubes (SWCNTs). We formulate the theory of the nuclear relaxation time in the framework of the Tomonaga-Luttinger liquid theory to enable the determination of the HFC from recent data by Ihara \\textit{et al.} [Ihara \\textit{et al.} EPL \\textbf{90}, 17004 (2010)]. 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