EtMe$_3$Sb[Pd(dmit)$_2$]$_2$, an organic Mott insulator with nearly isotropic triangular lattice, is a candidate material for a quantum spin liquid, in which the zero-point fluctuations do not allow the spins to order. The itinerant gapless excitations inferred from the thermal transport measurements in this system have been a hotly debated issue recently. While the presence of a finite linear residual thermal conductivity, $kappa_0/T equiv kappa/T (T rightarrow 0)$, has been shown [M. Yamashita {it et al.} Science {bf 328}, 1246 (2010)], recent experiments [P. Bourgeois-Hope {it et al.}, Phys. Rev. X {bf 9}, 041051 (2019); J. M. Ni {it et al.}, Phys. Rev. Lett. {bf 123}, 247204 (2019)] have reported the absence of $kappa_0/T$. Here we show that the low-temperature thermal conductivity strongly depends on the cooling process of the sample. When cooling down very slowly, a sizable $kappa_0/T$ is observed. In contrast, when cooling down rapidly, $kappa_0/T$ vanishes and, in addition, the phonon thermal conductivity is strongly suppressed. These results suggest that possible random scatterers introduced during the cooling process are responsible for the apparent discrepancy of the thermal conductivity data in this organic system. The present results provide evidence that the true ground state of EtMe$_3$Sb[Pd(dmit)$_2$]$_2$ is likely to be a quantum spin liquid with itinerant gapless excitations.
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