Stability of Matter: $E(Z)\sim -Z^2$

This is a follow up of previous posts on the basic problem of Stability of Matter.

The Lieb-Thirring/Dyson-Lenard proof of the scaling $E(Z) \sim -Z^{\frac{7}{3}}$ of the ground state energy $E(Z)$ of an atom with kernel charge $Z$, is viewed to be a master-piece of rigorous mathematics revealing a deep truth of quantum mechanics. 

But empirical observations of total energies of atoms show a slower scaling of $-Z^\alpha$ with $\alpha \approx 1.81$ according to the following compilation by chatGPT:

The Lieb-Thirring proof is based on the Thomas-Fermi model and not the Schrödinger equation of Standard Quantum Mechanics StdQM, suggesting that the Thomas-Fermi model does not capture physics. But there is no theoretical proof based on StdQM as the canonical model. 

Let us see what RealQM can deliver. We then recall the scaling emerging from RealQM computations of atoms in spherical symmetry recorded in this post showing $E(Z)\sim -log(Z)Z^2$ thus essentially $E(Z)\sim -Z^2$ in much better agreement with observation than the Lieb-Thirring result $-Z^{\frac{7}{3}}$.

A heuristic proof of the RealQM estimate $E(Z)\sim -log(Z)Z^2$ is given in the post starting from an observation that the charge density roughly scales with $\frac{Z}{r^2}$ with $r$ distance to the kernel.