The solitary wave conjecture for focusing mass-critical NLS

Let $d\ge 1$, and consider the focusing mass-critical nonlinear Schrödinger equation. Let $Q$ denote its ground state, and suppose that a global solution with initial data $u_0\in L_x^2(\mathbb{R}^d)$ has ground-state mass $M(u_0)=M(Q)$. A solution is non-scattering if it does not scatter in the relevant time direction, and the solitary wave is the solution generated by $Q$. Solitary wave conjecture. Every global non-scattering solution with ground-state mass is the solitary wave up to the translation and Galilean symmetries of the equation. These are conjectured to be the only global obstruction to scattering at ground-state mass, alongside the pseudoconformal finite-time blowup solution.