The spin-lattice relaxation rate $T_1^{-1}$ of $^1$H-NMR has been measured in (CH$_3$)$_2$CHNH$_3$Cu(Cl$_x$Br$_{1-x}$)$_3$ with $x=0$ and 0.35, in order to investigate the microscopic magnetism of systems. Previous macroscopic magnetization and specific heat measurements suggested that these two exist in a singlet-dimer phase. The temperature dependence of $T_1^{-1}$ in an $x=0$ system decreased exponentially toward zero, indicating microscopic evidence of the gapped singlet ground state, which is consistent with the macroscopic experiments. At the same time, in the $x=0.35$ system, $T_1^{-1}$ showed a sharp peak structure at around 7.5 K though no splitting of $^1$H-NMR spectra indicative of the magnetic ordering was observed. We discuss the observed sharp peak structure in the $x=0.35$ system with the soft mode toward the exotic magnetic ground state suggested by the recent $mu$SR experiments.
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