Rigidity conjecture at the ground-state mass for mass-critical NLS

Let $u$ solve the focusing mass-critical nonlinear Schrödinger equation with initial data $u(0)=u_0\in L_x^2(\mathbb{R}^d)$, and let $M$ denote the conserved mass. Let $Q$ be the unique positive radial Schwartz solution of

$$\Delta Q-Q+|Q|^{\frac 4d}Q=0.$$

Define the solitary wave and pseudo-conformal ground state by

$$SW=e^{it}Q(x),\qquad Pc(Q)=|t|^{-\frac d2}e^{\frac{i|x|^2-4}{4t}}Q\left(\frac{x}{t}\right).$$

Rigidity conjecture at the ground-state mass. If $M(u_0)=M(Q)$, then only the following cases can occur: if $u$ blows up in finite time, it coincides with $Pc(Q)$ up to symmetries of the equation; if $u$ is global, then it either scatters in both time directions or coincides with $SW$ up to symmetries of the equation.

This conjecture seeks a complete classification of minimal-mass non-scattering solutions. The explicitly known threshold examples are the solitary wave and pseudo-conformal solution, but the asserted classification remains open in the source.