Critical norm conjecture for defocusing inter-critical nonlinear Schrödinger equations

Consider the defocusing Schrödinger initial value problem for $u:I\times\mathbb{R}^{d}\to\mathbb{C}$,

$$(i\partial_t+\Delta_{\mathbb{R}^{d}})u=F(u)=|u|^{p}u,\qquad u(0,x)=u_0\in\dot{H}^{s_c}(\mathbb{R}^{d}),$$

where $s_c\geq 0$ and $p=\frac{4}{d-2s_c}$. The critical norm conjecture asserts that if

$$\sup_{t\in I}\|u(t)\|_{\dot{H}^{s_c}(\mathbb{R}^{d})}<\infty,$$

where $I$ is the lifespan of $u$, then the solution is globally well-posed and scatters. This conjecture formulates the expected continuation and scattering criterion for the defocusing inter-critical nonlinear Schrödinger equation; the paper studies scattering in a high-dimensional inter-critical regime, but the supplied text does not establish the conjecture in the full stated generality.