Soliton resolution conjecture for bounded energy-critical wave solutions

Consider a bounded solution $u$ of the focusing energy-critical nonlinear wave equation (NLW). A soliton resolution conjecture predicts that, as time approaches the maximal time of existence, $u$ asymptotically decouples into a finite sum of modulated solitons, together with a regular part in the finite-time blow-up case or free radiation in the global case, and a residue term vanishing in the energy space. It is also conjectured that every bounded solution is a sum of modulated, decoupled traveling waves and a scattering part. This conjecture describes the expected asymptotic dynamics of bounded energy solutions; the precise general decomposition remains open.