The electrodynamic response of the organic spin-liquid candidate $kappa$-(BEDT-TTF)$_2$Cu$_2$(CN)$_3$ has been measured in an extremely wide energy range ($10^{-13}$ to 2 eV) as a function of temperature (5 to 300 K). Below the Mott gap, excitations from the un-gapped spinon continuum cause a considerable contribution to the infrared conductivity, as suggested by the U(1) gauge theory. At THz frequencies we can identify a power-law behavior $sigma(omega) propto omega^{beta}$ with two distinct exponents $beta$ that change from 0.9 to 1.3 at low temperatures. The corresponding crossover scales with temperature: $hbaromega_c approx k_B T$. The observed exponents differ by more than a factor of 2 from the theoretically predicted ones. The findings are compared with those obtained on Herbertsmithites.
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