  The hyperkagome antiferromagnet Na$_{4}$Ir$_3$O$_8$ represents the first
genuine candidate for the realisation of a three-dimensional quantum
spin-liquid. It can also be doped towards a metallic state, thus offering a
rare opportunity to explore the nature of the metal-insulator transition in
correlated, frustrated magnets. Here we report thermodynamic and transport
measurements in both metallic and weakly insulating single crystals down to 150
mK. While in the metallic sample the phonon thermal conductivity
($\kappa^{ph}$) is almost in the boundary scattering regime, in the insulating
sample we find a large reduction $\kappa^{ph}$ over a very wide temperature
range. This result can be ascribed to the scattering of phonons off nanoscale
disorder or off the gapless magnetic excitations that are seen in the
low-temperature specific heat. This works highlights the peculiarity of the
metal-insulator transition in Na$_{3+x}$Ir$_3$O$_8$ and demonstrates the
importance of the coupling between lattice and spin degrees of freedom in the
presence of strong spin-orbit coupling.

Strongly Correlated Electrons