The critical-norm scattering conjecture for defocusing NLS in Euclidean space

Consider the defocusing nonlinear Schrödinger equation

$$iu_t+\Delta u=|u|^{\alpha}u$$

in $\mathbb{R}^d$, with $d\geq 1$ and $\alpha\geq 4/d$. Set $s_c=d/2-2/\alpha$, and let $u:I\times\mathbb{R}^d\to\mathbb{C}$ be a maximal-lifespan solution. Critical-norm scattering conjecture. If

$$u\in L_t^\infty \dot{H}_x^{s_c}(I\times\mathbb{R}^d),$$

then $u$ is global and scatters as $t\to\pm\infty$. This is the conjectural global well-posedness and scattering principle under a priori critical-norm control; many mass- and energy-critical cases and several other ranges are known, but the full statement remains open.