  We study the propagation of a density perturbation in a weakly interacting
boson gas confined on a lattice and in the presence of square dimerized
impurities. Such a two-dimensional random-dimer model (2D-DRDM), previously
introduced in [Capuzzi et al., Phys. Rev. A 92, 053622 (2015)], is the disorder
transition from a single square lattice, where impurities are absent, to a
bipartite square lattice, where the number of impurities is maximum and
coincides with half the number of lattice sites. We show that disorder
correlations can play a crucial role in the dynamics for a broad range of
parameters by allowing density fluctuations to propagate in the 2D-DRDM
lattice, even in the limit of strong disorder. In such a regime, the
propagation speed depends on the percentage of impurities, interpolating
between the speed in a single monoperiodic lattice and that in a bipartite one.
