Can the Hartree-Fock kinetic energy exceed the exact kinetic energy?

The Hartree-Fock (HF) approximation has been an important tool for quantum-chemical calculations since its earliest appearance in the late 1920s and remains the starting point of most single-reference methods in use today. Intuition suggests that the HF kinetic energy should not exceed the exact kinetic energy; but no proof of this conjecture exists, despite a near century of development. Beginning from a generalized virial theorem derived from scaling considerations, we derive a general expression for the kinetic energy difference that applies to all systems. For any atom or ion, this trivially reduces to the well-known result that the total energy is the negative of the kinetic energy and, since correlation energies are never positive, proves the conjecture in this case. Similar considerations apply to molecules at their equilibrium bond lengths. We use highly precise calculations on Hooke's atom (two electrons in a parabolic well) to test the conjecture in a nontrivial case and to parameterize the difference between density functional and HF quantities, but find no violations of the conjecture.