Nonexistence of below-ground-state almost periodic solutions for focusing generalized KdV

Let $u$ be an almost periodic solution modulo symmetries of the focusing, mass-critical generalized KdV equation $(1.1)$, meaning that there are continuous functions $N(t):I\to(0,\infty)$ and $x(t):I\to\mathbb{R}$ such that

$$\{N(t)^{-1/2}u(t,N(t)^{-1}x+x(t)):t\in I\}$$

is contained in a compact subset of $L^2(\mathbb{R})$. Let $Q$ be the positive ground state defined by

$$Q(x)=\frac{3^{1/4}}{\cosh^{1/2}(2x)}.$$

Below-ground-state almost-periodic-solution conjecture. There does not exist a nonzero almost periodic solution $u$ to $(1.1)$ satisfying

$$0<\|u\|_{L^2}<\|Q\|_{L^2}.$$

This is a concentration-compactness formulation of the expected scattering theory. It would follow from the below-ground-state global well-posedness and scattering conjecture, but the paper notes that it is not by itself sufficient to obtain that conjecture in the full mass range; the statement remains open.