Spacetime bounds for the mass-critical nonlinear Schrödinger equation

Let $v$ solve the mass-critical nonlinear Schrödinger equation on ${\mathbb R}\times{\mathbb R}$, let $v_0\in L^2_x({\mathbb R})$ be its initial data, let $M(v_0)$ denote its mass, and let $Q$ be the ground state. In the focusing case assume $M(v_0)<2\sqrt{\frac65}M(Q)$. Spacetime-bounds conjecture. The defocusing mass-critical nonlinear Schrödinger equation is globally well-posed for arbitrary initial data $v_0\in L^2_x({\mathbb R})$. In the focusing case, the same conclusion holds for initial data with $M(v_0)<2\sqrt{\frac65}M(Q)$. In both cases, the global solution satisfies

$$\int_{\mathbb R}\int_{\mathbb R}|v(t,x)|^6\,dx\,dt\leq C(M(v_0)).$$

The source explicitly states that this is the natural global well-posedness and scattering conjecture and that it remains open. The spacetime bound is the critical global estimate encoding both properties.