  We test the analytical formula for the enhancement of the nuclear magnetic
resonance rate $T_1^{-1}$ by the critical spin fluctuations, over the simple
power-law dependence predicted for a purely one-dimensional spin system,
recently derived in the random phase approximation [M. Dupont et al., Phys.
Rev. B 98, 094403 (2018)]. This prediction is experimentally confirmed by
excellent fits to the published temperature dependence of $T_1^{-1}$ data in
the two representative spin compounds, (C$_7$H$_{10}$N)$_2$CuBr$_4$ (DIMPY) and
BaCo$_2$V$_2$O$_8$, providing at the same time a direct and convenient
experimental determination of the Tomonaga-Luttinger-liquid parameter $K$, very
well in agreement with theoretical predictions.
