The known asymptotic behavior of the total energy of two weakly interacting systems imposes stringent conditions on the exchange-correlation energy as a functional of the one-electron reduced density matrix. Although the first-order conditions that involve Coulomb-type two-electron integrals are relatively trivial to satisfy, the exact functional should also conform to two second-order expressions, and consequently to certain sum rules. The primitive natural spin-orbital functionals satisfy the first-order conditions but, lacking terms quadratic in two-electron integrals, are found to be incapable of recovering the dispersion component of the interaction energy. Violating the sum rules, the recently proposed Yasuda functional yields nonvanishing dispersion energy with spurious asymptotic terms that scale like inverse fourth and fifth powers of the intersystem distance.