The $L^2$-critical mBO global-existence and blow-up dichotomy

Let $u_0\in H^{1/2}(\mathbb R)$ be sufficiently localized, let $Q$ be the ground state solution of the equation defining $Q$ with $m=3$, and let $u(x,t)$ be the mBO time evolution of $u_0$. The energy is denoted by $E[u_0]$. The $L^2$-critical mBO dichotomy. The solution satisfies both:

1. $u(x,t)$ exists globally in time if $\\|u_0\\|_{L^2}<\\|Q\\|_{L^2}$;
2. $u(x,t)$ blows up in finite time if $E[u_0]<0$, which implies $\\|u_0\\|_{L^2}>\\|Q\\|_{L^2}$.

The first assertion is known by the argument of Weinstein, while positive-energy blow-up solutions also exist above the soliton mass. The conjectural part concerns the stated finite-time blow-up criterion and the resulting critical-mass dynamics.