Mittleman's conjecture on the Dirac–Fock ground state energy

Let $e_q$ denote the physical ground state energy of the electron-positron Hartree–Fock model, obtained by maximizing the model's ground state energy over all allowed one-particle electron subspaces, and let $E_q$ denote the ground state energy of the Dirac–Fock model. Mittleman's conjecture. For $\alpha$ small and $c$ large, $E_q=e_q$. The conjecture would justify the Dirac–Fock model as an approximation to the electron-positron Hartree–Fock model, and hence as a rigorous approximation to quantum electrodynamics when vacuum polarization is neglected. The source states that the conjecture is proved when $q=1$ or when $\sigma_q(D-V)<\sigma_{q+1}(D-V)$, but may fail in other cases.