Below-ground-state scattering conjecture for the focusing mass-critical generalized KdV equation

Let $u$ solve the focusing, mass-critical generalized KdV equation $(1.1)$ with initial data $u(0,x)=u_0\in L^2(\mathbb{R})$, and let

$$Q(x)=\frac{3^{1/4}}{\cosh^{1/2}(2x)}>0$$

be the ground state, which satisfies $Q_{xx}+Q^5=Q$. Below-ground-state scattering conjecture. If

$$\|u_0\|_{L^2}<\|Q\|_{L^2},$$

then the solution to $(1.1)$ is globally well-posed and scattering. This is the generalized KdV analogue of the corresponding result known for the focusing, mass-critical nonlinear Schrödinger equation; the conjecture remains open.