  The spin-liquid candidate $\kappa$-(ET)$_2$Cu$_2$(CN)$_3$ [ET:
bis(ethylenedithio)tetrathiafulvalene] does not exhibit magnetic ordering down
to a very low temperature, but shows a mysterious anomaly at 6 K. The origin of
the so-called 6 K anomaly is still under debate. We carried out nuclear
quadrupole resonance (NQR) measurements on the copper sites of the insulating
layers, which are sensitive to the charge dynamics unlike the conventional
spin-1/2 nuclear magnetic resonance (NMR). The main finding of this study is
that the observation of a sharp peak behavior in the nuclear spin-lattice
relaxation rate $T_1^{-1}$ of $^{63}$Cu NQR at 6 K while $T_1^{-1}$ of both
$^{13}$C and $^{1}$H NMR show no clear anomaly. This behavior can be understood
as a second-order phase transition related to charge disproportionation in the
ET layers.
