Let X = G/K be a symmetric space of noncompact type, F a Zariski-dense subgroup of G with critical exponent 3(F). We show that all F-invariant conformal densities of dimension 6(F) (e.g. Patterson-Sullivan densities) have their support contained in a same and single G-orbit on the geometric boundary of X. In the lattice case, we explicitly determine 6(F) and this G-orbit, and we establish the uniqueness of such densities.