The "fixed diagonal matrices" (FDM) dispersion formalism [Kooi, D. P.; et al. J. Phys. Chem. Lett. 2019, 10, 1537] is based on a supramolecular wave function constrained to leave the diagonal of the many-body density matrix of each monomer unchanged, reducing dispersion to a balance between kinetic energy and monomer-monomer interaction. The corresponding variational optimization leads to expressions for the dispersion energy in terms of the ground-state pair densities of the isolated monomers only, providing a framework to build new approximations without the need for polarizabilities or virtual orbitals. Despite the underlying microscopic real space mechanism being incorrect, as in the exact case there is density relaxation, the formalism has been shown to give extremely accurate (or even exact) dispersion coefficients for H and He. The question we answer in this work is how accurate the FDM expressions can be for isotropic and anisotropic C6 dispersion coefficients when monomer pair densities are used from different levels of theory, namely Hartree-Fock, MP2, and CCSD. For closed-shell systems, FDM with CCSD monomer pair densities yield a mean average percent error for isotropic C6 dispersion coefficients of about 7% and a maximum absolute error within 18%, with a similar accuracy for anisotropies. The performance for open-shell systems is less satisfactory, with CCSD pair densities performing sometimes worse than Hartree-Fock or MP2. In the present implementation, the computational cost on top of the monomer's ground-state calculations is O(N4). The results show little sensitivity to the basis set used in the monomer's calculations.