  The thermal conductivity $\kappa$ of the quasi-2D organic spin-liquid
candidate EtMe$_3$Sb[Pd(dmit)$_2$]$_2$ (dmit-131) was measured at low
temperatures, down to 0.07 K. We observe a vanishingly small residual linear
term $\kappa_0/T$, in $\kappa/T$ vs $T$ as $T \to 0$. This shows that the
low-energy excitations responsible for the sizeable residual linear term
$\gamma$ in the specific heat $C$, seen in $C/T$ vs $T$ as $T \to 0,$ are
localized. We conclude that there are no mobile gapless excitations in this
spin liquid candidate, in contrast with a prior study of dmit-131 that reported
a large $\kappa_0/T$ value [Yamashita et al., Science 328, 1246 (2010)]. Our
study shows that dmit-131 is in fact similar to
$\kappa$-(BEDT-TTF)$_2$Cu$_2$(CN)$_3$, another quasi-2D organic spin-liquid
candidate where a vanishingly small $\kappa_0/T$ and a sizeable $\gamma$ are
seen. We attribute heat conduction in these organic insulators without magnetic
order to phonons undergoing strong spin-phonon scattering, as observed in
several other spin-liquid materials.
