Codimension-one threshold manifold conjecture for the cubic wave equation

Let $\mathcal{H}^s=\dot H^s(\mathbb{R}^3)\times\dot H^{s-1}(\mathbb{R}^3)$ denote the phase space, and consider initial data for the cubic wave equation. The self-similar profile in this conjecture is $S(t)=\sqrt{2}/t$. Codimension-one threshold manifold conjecture. The set of initial data in $\mathcal{H}^1\cap\mathcal{H}^{1/2}$ whose solutions satisfy the asymptotic relation above with $S(t)=\sqrt{2}/t$ is a codimension-one submanifold that separates scattering from ODE blow-up. The source presents this as an expected structure analogous to a previously constructed manifold, with no proof supplied.