Do we know the asymptotic behavior of density functional in DFT?

I was reading a book on DFT the other day, and the author included asymptotic behavior for $E\_{xc}[\rho(r)]$. 


Given that, do we know the asymptotic behavior for kinetic energy and/or total energy in DFT?


A very important advice will be to read again the principles of density functional theory to better understand the approximations of $E\_{xc}[\rho(r)]$ from LDA to GGA climbing gradually the Jacobs ladder.


There is no expression of the kinetic energy (KE) depending on the density $T[\rho(r)]$ in DFT, however there are tricks to deduce KE after solving the Kohn-Sham's equation.
DFT has generally no problem at small distances (interatomic) where the density tends to be homogeneous, even a crude approximation like LDA can handle this situation.


At large distances, the density is more difficult to handle for DFT (inhomogenious, or atomic-like) even by using corrections like GGA. A non-local approximation like Hartree-Fock (HF) is more suitable for the situation. If you plot the dissociation energy or the interatomic potential (Total energy vs interatomic distance) by using a simple DFT functional compared to HF you will notice this huge difference for $r>>$, the asymptotic behaviour of DFT to the dissociation energy is pathological.


The main drawback of DFT is the fact that eigenvalues are upshifted, called self-interaction. This pathological asymptotic behavior is due to this self-interaction (separating DFT and HF). The intuitive correction is to move towards HF. Therefore, there are hybrid functionals integrating HF exact exchange. Until now, major corrections are range-separated functionals.


Obviously, all the corrections affect only $E\_{xc}[\rho(r)]$ not the kinetic energy consequently the total energy.


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