Scattering conjecture for critical gKdV

Consider the generalized Korteweg–de Vries equation with nonlinearity exponent $p=5$ and sign $\mu=\pm1$. Let $M(u)$ denote the conserved mass and let $Q$ be the ground state. For a compact time interval $I$, let $u:I\times\mathbf{R}\to\mathbf{R}$ be a Schwartz solution, meaning that $u$ is smooth and all spacetime derivatives of $u(t)$ are Schwartz in space, locally uniformly in time.

Scattering conjecture for critical gKdV. There exist functions

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

such that

$$\|u\|_{L^5_xL^{10}_t(I\times\mathbf{R})}\leq f_\mu(M(u)).$$

This conjecture would give strong global control and scattering for energy-class solutions in the defocusing case, or in the focusing case below the ground-state mass. It is known for sufficiently small mass, but was not known in general in the source.