Logarithmic critical-norm criterion for global scattering of cubic-wave solutions

Let $u$ be a solution of the cubic wave equation on its maximal interval $[0,t_+)$, with initial data in $\mathcal{H}^{1/2}$. Logarithmic critical-norm criterion. There exists $\beta>0$ such that, if

$$\sup_{2\leq t<t_+}\|\boldsymbol{u}(t)\|_{\mathcal{H}^{1/2}}(\log t)^{-\beta}<\infty,$$

then $t_+=\infty$ and $u$ scatters to a linear solution. This would give a logarithmic-growth threshold preventing finite-time blow-up and forcing scattering; the source presents it as conjectural and supplies no proof.