We scrutinize the magnetic properties of $kappa$-(BEDT-TTF)$_2$Hg(SCN)$_2$Cl through its first-order metal-insulator transition at $T_{rm CO}=30$ K by means of $^1$H nuclear magnetic resonance (NMR). While in the metal we find Fermi-liquid behavior with temperature-independent $(T_1T)^{-1}$, the relaxation rate exhibits a pronounced enhancement when charge order sets in. The NMR spectra remain unchanged through the transition and no magnetic order stabilizes down to 25 mK. Similar to the isostructural spin-liquid candidates $kappa$-(BEDT-TTF)$_2$Cu$_2$(CN)$_3$ and $kappa$-(BEDT-TTF)$_2$Ag$_2$(CN)$_3$, $T_1^{-1}$ acquires a dominant maximum (here around 5 K). Field-dependent experiments identify the low-temperature feature as a dynamic inhomogeneity contribution that is typically dominant over the intrinsic relaxation but gets suppressed with magnetic field.
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