The scattering conjecture for mass-critical NLS

Let $u$ solve the mass-critical nonlinear Schrödinger equation on ${\mathbf{R}}^d$, let $M(u)$ denote its mass, and let $S(u)$ denote its scattering size. Define

$$m_0:=\inf\{m\geq 0:A(m)=+\infty\},$$

where $A(m)$ is the supremum of $S(u)$ over solutions with $M(u)\leq m$. In the focusing case, let $Q$ be the ground state satisfying

$$\Delta Q+Q^{1+4/d}=Q.$$

Scattering conjecture. In the defocusing case ($\mu=+1$) we have $m_0=+\infty$, while in the focusing case ($\mu=-1$) we have $m_0=M(Q)$. This predicts global well-posedness and scattering for every defocusing $L^2$ datum and identifies the ground-state mass as the sharp threshold in the focusing problem. The statement is presented as a conjecture in the source; its resolution is not specified in the supplied material.