Conjectured energy-critical Type II singularity formation

Let $\mathcal{X}=(\{v\leq 0\}\cup \{t\leq -0.9\})\cap\{-1\leq u<0\}$ and let $\mathcal{C}$ be the backward light cone. The symbols $\phi$, $r$, $\nu$, $d$, $p$, and $k$ have the meanings in the corresponding wave-map or semilinear equation, and $C^{s-}$ denotes regularity strictly below $C^s$; infinite conormal regularity is regularity to all orders with respect to the vector fields tangent to the relevant characteristic structure. Energy-critical Type II conjecture. The listed Type II solutions should exist in $\mathcal{X}$: for any $\nu>1$, the three solutions for $(d,k)=(2,1)$ wave maps, $(d,p)=(3,5)$, and $(d,p)=(4,3)$ have respectively the stated bounds $\phi|_{\mathcal{C}}=\mathcal{O}(r^{\nu-1})$, $\mathcal{O}(r^{(\nu-2)/2})$, and $\mathcal{O}(r^{\nu-2})$, with the stated $C^{\nu-1/2-}$ or $C^{\nu/2-}$ regularity and infinite conormal regularity across $\mathcal{C}$; for $\nu\in\mathbb{N}_{\geq2}$ there is a smooth wave-map solution with $(d,k)=(2,1)$ and $\phi|_{\mathcal{C}}=\mathcal{O}(r^\nu)$; and there are smooth solutions with the three additional decay rates $\mathcal{O}(e^{-\sqrt{|\log r|}})$, $\mathcal{O}(|\log r|^{-1/(2k-2)})$ for $d=2$, $k\geq2$, and $\mathcal{O}(r^{-1}e^{-\sqrt{|\log r|}+O(1)})$ for $(d,p)=(4,3)$. In all cases the solutions are smooth for $v<0$, have scale-critical norm bounded away from zero as $\tau\to0$, and the stated smoothness and conormality may be replaced by finite $C^k$ versions. These conjectures organise known Type II constructions according to their exterior regularity and decay, while the cited results do not uniformly record the precise decay or high $r\partial_x,r\partial_t$ regularity across $\mathcal{C}$; the claims therefore remain open in the supplied source context.