Quasi-one-dimensional (1D) antiferromagnets are good candidates to realize a Tomonaga-Luttinger liquid (TLL), describing 1D interacting quantum systems, provided 3D coupling remains small as compared to temperature. In systems such as NiCl-4SC(NH) (DTN), made of weakly coupled spin chains, interchain coupling plays an important role at low temperature as it leads to antiferromagnetic long-range order below K, corresponding to a Bose-Einstein condensate of magnons. Using time-dependent numerical simulations at finite temperature, the authors compute here the nuclear magnetic resonance (NMR) relaxation rate , a key experimental quantity, and address the question of the low-temperature crossover between TLL predictions and the higher-temperature regime in purely D quantum spins chains. They also comment on the experimental observation of the TLL regime, possibly masked by D long-range order in materials such as DTN.