Using linear flavor-wave theory (LFWT) and auxiliary field quantum Monte Carlo (QMC), we investigate the properties of the SU(4) Heisenberg model on the anisotropic square lattice in the fully antisymmetric six-dimensional irreducible representation, a model that describes interacting fermions with four flavors at half-filling. Thanks to the calculations on very large systems, we have been able to convincingly demonstrate that QMC results are consistent with a small but finite antiferromagnetic moment at the isotropic point, in qualitative agreement with LFWT obtained earlier [F. H. Kim et al., Phys. Rev. B 96, 205142 (2017)], and in quantitative agreement with results obtained previously on the Hubbard model [D. Wang et al., Phys. Rev. Lett. 112, 156403 (2014)] after extrapolation to infinite U/t. The presence of a long-range antiferromagnetic order has been further confirmed by showing that a phase transition takes place into a valence-bond solid (VBS) phase not too far from the isotropic point when reducing the coupling constant along one direction on the way to decoupled chains.