Dynamical neutrality conjecture for Thomas–Fermi theory

Let $c_0>0$ be fixed, and consider the time-dependent Thomas–Fermi system

$$\begin{cases} \partial_t\varphi=\frac{1}{2}(\nabla\varphi)^2+c_0\rho^{2/3}-Z|x|^{-1}+\rho*|x|^{-1},\\ \partial_t\rho=\nabla(\rho\nabla\varphi). \end{cases}$$

Thomas–Fermi dynamical neutrality conjecture. For every initial state $(\varphi_0,\rho_0)$ satisfying $0\leqslant\rho_0\in L^1(\mathbb{R}^3)\cap L^{5/3}(\mathbb{R}^3)$ and $\sqrt{\rho_0}|\nabla\varphi_0|\in L^2(\mathbb{R}^3)$, and for every $R>0$,

$$\limsup_{T\to\infty}\frac{1}{T}\int_0^T\int_{|x|\leqslant R}\rho(x,t)\,\mathrm{d}x\,\mathrm{d}t\leqslant Z.$$

This is the Thomas–Fermi analogue of the dynamical Hartree conjecture and is presented as open.