  The Enskog kinetic theory of multicomponent granular suspensions employed
previously [G\'omez Gonz\'alez, Khalil, and Garz\'o, Phys. Rev. E \textbf{101},
012904 (2020)] is considered further to determine the four transport
coefficients associated with the heat flux. These transport coefficients are
obtained by solving the Enskog equation by means of the application of the
Chapman--Enskog method around the local version of the homogeneous state.
Explicit forms of the heat flux transport coefficients are provided in
steady-state conditions by considering the so-called second Sonine
approximation to the distribution function of each species. Their quantitative
variation on the control parameters of the mixture (masses and diameters,
coefficients of restitution, concentration, volume fraction, and the background
temperature) is demonstrated and the results show that in general the
dependence of the heat flux transport coefficients on inelasticity is clearly
different from that found in the absence of the gas phase (\emph{dry} granular
mixtures). As an application of the general results, the stability of the
homogeneous steady state is analyzed by solving the linearized Navier--Stokes
hydrodynamic equations. The linear stability analysis (which holds for
wavelengths long compared with the mean free path) shows that the transversal
and longitudinal modes are always stable with respect to long-enough wavelength
excitations. This conclusion agrees with previous results derived for
monocomponent and (dilute) bidisperse granular suspensions but contrasts with
the instabilities found in previous works in dry (no gas phase) granular
mixtures.
