Mass-critical cubic NLS scattering conjecture

Let $\mu=\pm1$, and let $w_0\in L^2_x(\mathbb{R}^2)$. In the focusing case assume $M(w_0)<M(Q)$. A solution scatters if there are $w_\pm\in L^2_x$ with $\|w(t)-e^{it\Delta}w_\pm\|_{L^2_x}\to0$ as $t\to\pm\infty$. NLS conjecture. There exists a unique global solution $w$ to

$$iw_t+\Delta w=\mu|w|^2w,$$

with $w(0)=w_0$, satisfying

$$\|w\|_{L^4_{t,x}(\mathbb{R}\times\mathbb{R}^2)}\leq C(M(w_0)).$$

Consequently, $w$ scatters in both time directions. The supplied text states that this conjecture has been proved in the defocusing case and for spherically symmetric data in the focusing case.