Energy-critical NLS without potential global well-posedness and scattering conjecture

Let $d\geq 3$, let $\mu\in\{1,-1\}$, and consider the energy-critical nonlinear Schrödinger equation without potential

$$(i\partial_t+\tfrac{1}{2}\Delta)u=\mu|u|^{\frac{4}{d-2}}u, \qquad u(0)=u_0\in\dot{H}^1(\mathbb{R}^d).$$

For $\mu=-1$, define

$$W(x)=\frac{1}{\left(1+\frac{2|x|^2}{d(d-2)}\right)^{\frac{d-2}{2}}}\in\dot{H}^1(\mathbb{R}^d),$$

where $W$ solves $\tfrac{1}{2}\Delta W+|W|^{\frac{4}{d-2}}W=0$. Global well-posedness and scattering conjecture. When $\mu=1$, solutions exist globally and scatter: for every $u_0\in\dot{H}^1(\mathbb{R}^d)$ there is a unique global solution satisfying

$$S_{\mathbb{R}}(u):=\int_{\mathbb{R}}\int_{\mathbb{R}^d}|u(t,x)|^{\frac{2(d+2)}{d-2}}\,dx\,dt\leq C(E_\Delta(u_0))<\infty,$$

and there exist $u_\pm\in\dot{H}^1(\mathbb{R}^d)$ such that

$$\lim_{t\to\pm\infty}\left\|u(t)-e^{\pm\frac{it\Delta}{2}}u_\pm\right\|_{\dot{H}^1}=0,$$

with the maps $u_0\mapsto u_\pm(u_0)$ homeomorphisms of $\dot{H}^1$. When $\mu=-1$, global well-posedness and scattering also hold provided

$$E_\Delta(u_0)<E_\Delta(W),\qquad \|\nabla u_0\|_{L^2}<\|\nabla W\|_{L^2}.$$

This conjecture is stated as background for the potential problem; the source later records that it is proved in the defocusing case, for radial data in the focusing case when $d\geq3$, and for all data in the focusing case when $d\geq5$.