  We study the nonequilibrium steady-state of a fully-coupled network of $N$
quantum harmonic oscillators, interacting with two thermal reservoirs. Given
the long-range nature of the couplings, we consider two setups: one in which
the number of particles coupled to the baths is fixed (intensive coupling) and
one in which it is proportional to the size $N$ (extensive coupling). In both
cases, we compute analytically the heat fluxes and the kinetic temperature
distributions using the nonequilibrium Green's function approach, both in the
classical and quantum regimes. In the large $N$ limit, we derive the asymptotic
expressions of both quantities as a function of $N$ and the temperature
difference between the baths. We discuss a peculiar feature of the model,
namely that the bulk temperature vanishes in the thermodynamic limit, due to a
decoupling of the dynamics of the inner part of the system from the baths. At
variance with usual cases, this implies that the steady state depends on the
initial state of the particles in the bulk. We also show that quantum effects
are relevant only below a characteristic temperature that vanishes as $1/N$. In
the quantum low-temperature regime the energy flux is proportional to the
universal quantum of thermal conductance.
