If so, how to see that? Also I think it has been proven that the total Chern-number for free fermion system is 0? If you know how to prove it, please make some comment or hopefully a sketch of proof. I encounter this in a funny way. In fact, I just want to confirm the maximum Chern-number on Honeycomb lattice is 2 which I conjectured from a very peculiar way.
(I know people have already developed method to design bands with arbitrary Chern number. However that does not violate the statement since what they've achieved is to have multilayered system which essentially have N orbits in one unit cell to have a band with Chern-number N)