Threshold transition conjecture for cubic focusing nonlinear waves

Consider the focusing nonlinear wave equation $P[\phi]=0$ with $p=3$ in spatial dimension $n\geq 5$. Let $(\phi_{0,\omega},\phi_{1,\omega})$, $\omega\in[0,1]$, be a one-parameter family of initial data imposed at $t=1$. Threshold transition conjecture. There exists a smooth one-parameter family $(\phi_{0,\omega},\phi_{1,\omega})$ and some $\omega^*\in(0,1)$ such that: (1) $(\phi_{0,0},\phi_{1,0})=(0,0)$, and for $0\leq\omega<\omega^*$ the corresponding solution exists globally in time to the past; (2) $(\phi_{0,1},\phi_{1,1})\equiv(c_3,-\alpha_3c_3)$ in some sufficiently large ball, and for $\omega^*<\omega\leq1$ the corresponding solution blows up to the past, with some portion of the past boundary of its maximal domain of existence being a smooth spacelike hypersurface $\Sigma^-_{\omega}$ along which the solution exhibits ODE-type blow-up; (3) at $\omega=\omega^*$, the past development blows up at $(t,x)=(0,0)$, and within the future light cone of this point the solution approaches the nontrivial self-similar solution as $t\downarrow0$, exhibiting locally naked singularity formation. This conjectures a threshold between the open small-data scattering regime and the large-data ODE-type blow-up regime, with the self-similar solution governing the transition.