Dynamical critical-mass conjecture for the Hartree equation

Let $t_c$ be the critical mass for the stationary Hartree theory. Consider the time-dependent Hartree equation

$$\mathbf{i}\,\partial_tu=\left(-\Delta-|x|^{-1}+|u|^2*|x|^{-1}\right)u.$$

Dynamical Hartree conjecture. For every initial state $u_0\in H^1(\mathbb{R}^3)$ and every $R>0$,

$$\limsup_{T\to\infty}\frac{1}{T}\int_0^T\int_{|x|\leqslant R}|u(x,t)|^2\,\mathrm{d}x\,\mathrm{d}t\leqslant t_c.$$

A weaker bound $4t_c$ was proved by Lenzmann and Lewin; the asserted sharp bound remains open.