Does an exact local exchange potential exist?

In Kohn-Sham density functional theory, equations for occupied orbital functions of a model state are derived from the exact ground-state energy functional of Hohenberg and Kohn. The exchange-correlation potential in these exact Kohn-Sham equations is commonly assumed to be a local potential function rather than a more general linear operator. This assumption is tested and shown to fail for the exchange potential in a Hartree-Fock model for atoms, for which accurate solutions are known.