Inter-critical NLS critical Sobolev boundedness conjecture

Let $d\geq 1$ and $p>0$ satisfy

$$s_c:=\tfrac{d}{2}-\tfrac{2}{p}\geq 0.$$

Let $u:I\times\mathbb{R}^d\to\mathbb{C}$ be a maximal-lifespan solution to the nonlinear Schrödinger equation, with $u\in L_t^\infty\dot{H}_x^{s_c}(I\times\mathbb{R}^d)$. Critical Sobolev boundedness conjecture. Then $u$ is global and scatters, and there is a function $C:[0,\infty)\to[0,\infty)$ such that

$$S_{\mathbb{R}}(u)\leq C\bigl(\|u\|_{L_t^\infty\dot{H}_x^{s_c}(\mathbb{R}\times\mathbb{R}^d)}\bigr).$$

This asserts that a uniform critical Sobolev bound prevents finite-time blowup and guarantees scattering, with the scattering size controlled only by that bound. The paper addresses cases of this statement in the inter-critical regime; the supplied status evidence indicates that the claim has been resolved.