Bounded critical Sobolev norm conjecture for defocusing NLS and NLW

For a defocusing nonlinear Schrödinger equation or nonlinear wave equation, let $u$ be a solution whose critical Sobolev norm remains bounded in time. Bounded critical Sobolev norm conjecture. Any such solution is global-in-time and scatters. This conjecture extends the global well-posedness and scattering theory beyond the mass- and energy-critical cases, where a priori bounds in the critical Sobolev space serve as a substitute for a missing conservation law at critical regularity. Its general validity remains open.