Ground state conjecture for energy-critical NLS

Let $d\geq 3$ and consider the focusing energy-critical initial-value problem with initial data $u_0\in\dot{H}^1(\mathbb{R}^d)$. Let $W$ be the radial ground state satisfying

$$\Delta_{\mathbb{R}^d}W=-|W|^{\frac{4}{d-2}}W,$$

and suppose that the solution $u(t)$ has lifespan $I$ and satisfies

$$\sup_{t\in I}\|u(t)\|_{\dot{H}^1(\mathbb{R}^d)}<\|W\|_{\dot{H}^1(\mathbb{R}^d)}.$$

Ground state conjecture. The solution extends uniquely to a global solution $u\in C(\mathbb{R}:\dot{H}^1(\mathbb{R}^d))$ of

$$(i\partial_t+\Delta_{\mathbb{R}^d})u=-u|u|^{\frac{4}{d-2}},\qquad u(0)=u_0,$$

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

$$\|u(t)-e^{it\Delta_{\mathbb{R}^d}}\phi^{\pm}\|_{\dot{H}^1}\longrightarrow 0\quad\text{as }t\longrightarrow\pm\infty.$$

This is the below-ground-state scattering assertion for focusing energy-critical NLS. It concerns global existence and scattering below the ground-state threshold; the supplied source does not indicate whether the conjecture has been resolved.