Can the exchange-correlation potential of density functional theory be expressed solely in terms of HOMO and LUMO properties?

In a recent Letter with Ayers, we have demonstrated that the idempotent Dirac density matrix generated by any local potential V(r) can be derived in terms of HOMO and LUMO properties. Subsequently we have pointed out that the exchange-only potential Vx(r) of density functional theory can be built from the Dirac matrix γ(r1, r2) and its functional derivative δγ(r1, r2)/δρ(r). By utilizing further the equation of motion of the density matrix, we propose an answer in the affirmative to the question posed in the title of this Letter.