What is the current state of research into $v$-representability?

Question

In their proof, Hohenberg and Kohn (Phys Rev 136 (1964) B864) established that the ground state density, $\rho_\text{gs}$, uniquely determines the Hamiltonian. This had the effect of establishing an implicit relationship between $\rho_\text{gs}$ and the external potential (e.g. external magnetic field, crystal field, etc.), $V$, as the form of the kinetic energy and particle-particle interaction energy functionals are universal since they are only functions of the density. This implicit relationship defines a set of densities which are called $v$-representable. What is surprising is that there are "a number of 'reasonable' looking densities that have been shown to be impossible to be the ground state density for any $V$." (Martin, p. 130) On the surface, this restriction looks like it would reduce the usefulness of density functional theory, but, in practice, that is not the case. (See the proof by Levy - PNAS 76 (1979) 6062, in particular.) However, research continues into the properties of the $v$-representable densities, and I was wondering if someone could provide a summary of that work.

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