Exchange density-functional gradient expansion

The coefficient γₓ of the first term in a gradient expansion of the Hartree-Fock (HF) density functional was calculated by Sham to first order in e². It is now known that γₓHF diverges if e² is included to all orders. It has recently been claimed that if the exchange energy is defined in terms of density-functional (DF) eigenfunctions, rather than HF eigenfunctions, not only is γₓDF first order in e² but also γₓDF=γSham. It is proven in this paper that, in fact, γₓDF=8/7γSham.