The global-attractor conjecture for the Hartree equation

Let $u(t)$) be the unique solution to the Hartree equation for some $u_0\in H^1(\mathbb{R}^3)$, and let $\mathscr{A}_Z$ denote the global attractor. Take any sequence of times $t_n\to\infty$ such that $u(t_n)\rightharpoonup u_*$ weakly in $H^1(\mathbb{R}^3)$. The global-attractor conjecture. Then $u_*\in\mathscr{A}_Z$. This is a weaker form of soliton resolution: it asserts that every weak asymptotic limit belongs to the global attractor, despite the possible long-range effects of the Coulomb potential. The source gives no evidence that this conjecture has been resolved.