Universality conjecture for many-body atomic densities

Let $\rho_Z$ be the one-body density of a ground state of the many-body atomic Hamiltonian with nuclear charge $Z$ and $N=N_c(Z)$ electrons. Universality conjecture. Along subsequences $Z=Z_n\to\infty$, the densities have a pointwise limit away from the nucleus: there is a function $\rho_\infty$ such that

$$\lim_{n\to\infty}\rho_{Z_n}(x)=\rho_\infty(x),\qquad x\ne0.$$

Universality of the corresponding Thomas–Fermi and TFW quantities is known, whereas this many-body Schrödinger analogue remains open.