  Recent transport measurements [Churchill \textit{et al.} Nat. 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)]. Though we find that
$1/T_1$ is orders of magnitude enhanced with respect to a Fermi-liquid
behavior, the HFC has its usual, small value. Then, we reexamine the
theoretical description used to extract the HFC from transport experiments and
show that similar features could be obtained with HFC-independent system
parameters.
