Spacetime bounds for the mass-critical generalized KdV equation

Let $u$ solve the mass-critical generalized KdV equation on ${\mathbb R}\times{\mathbb R}$, let $u_0\in L^2_x({\mathbb R})$ be its initial data, let $M(u_0)$ denote its mass, and let $Q$ be the ground state. In the focusing case assume $M(u_0)<M(Q)$. Spacetime-bounds conjecture. The defocusing mass-critical generalized KdV is globally well-posed for arbitrary initial data $u_0\in L^2_x({\mathbb R})$. In the focusing case, the same conclusion holds for initial data with $M(u_0)<M(Q)$. In both cases, the global solution satisfies

$$\|u\|_{L_x^5L_t^{10}({\mathbb R}\times{\mathbb R})}\leq C(M(u_0)).$$

These bounds would establish global well-posedness and scattering at the mass-critical regularity, and would also imply strong stability. The supplied status is unknown, so the conjecture remains open in the database.