Scattering conjecture for critical NLS

Consider the $L^2$-critical nonlinear Schrödinger equation

$$-i\partial_tu+\partial_{xx}u=\mu|u|^{p-1}u,$$

with $p=5$ and $\mu=\pm1$, where $u:I\times\mathbf{R}\to\mathbf{C}$ is a Schwartz solution on a compact time interval $I$. Let $M(u)$ denote the conserved mass and let $Q$ be the ground state.

Scattering conjecture for critical NLS. There exist functions

$$f_+:[0,+\infty)\to[0,+\infty),\qquad f_-:[0,M(Q))\to[0,+\infty)$$

such that

$$\|u\|_{L^6_{t,x}(I\times\mathbf{R})}\leq f_\mu(M(u)).$$

This is the nonlinear analogue of the global Strichartz estimate for the free Schrödinger equation and would imply global control and scattering below the ground-state mass in the focusing case. The source presents the conjecture as open and notes that its small-mass case follows from the well-posedness theory.