The asymptotic electron-number and kinetic-energy conjecture for the Hartree equation

Let $u(t)$ be the unique solution to the Hartree equation for some $u_0\in H^1(\mathbb{R}^3)$, let $\mathscr{A}_Z$ be the global attractor, and let $\gamma_c$ be the constant appearing in the attractor bounds. Then, for all $r>0$, the asymptotic electron-number and kinetic-energy conjecture.

$$\limsup_{t\to\infty}\int_{|x|\leqslant r}|u(t,x)|^2\,dx\leqslant\sup_{u\in\mathscr{A}_Z}\int_{\mathbb{R}^3}|u|^2=\gamma_c Z$$

and

$$\limsup_{t\to\infty}\int_{|x|\leqslant r}|\nabla u(t,x)|^2\,dx\leqslant\sup_{u\in\mathscr{A}_Z}\int_{\mathbb{R}^3}|\nabla u|^2\leqslant\gamma_c Z^3.$$

Physically, regardless of the initial number of electrons and kinetic energy, at most $\gamma_c Z$ electrons should remain localized, with universally bounded total kinetic energy. The source notes that the kinetic-energy bound $\gamma_c Z^3$ is not optimal and gives no evidence that the conjecture has been resolved.