Conjectured energy-supercritical 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. Let $\phi$ solve the relevant wave-map or semilinear equation, with $d$ the spatial dimension, $p$ the power, and $c_{p,d}$ the constant determined by

$$\frac{p-1}{2}c_{p,d}^{p-1}=d-2-\frac{2}{p-1}.$$

Let $\gamma=\frac12(d-2-\sqrt{d^2-8d+8})$. Energy-supercritical Type II conjecture. For any $l\in\mathbb{N}_{>\gamma}$ and $d\geq7$, there exists a wave-map solution with $k=1$ such that $\phi|_{\mathcal{C}}=\frac\pi2+\mathcal{O}(r^{l-\gamma})$ and $\phi$ is smooth across $\mathcal{C}$. Moreover, for $d\geq11$ and $p>1+\frac4{d-4-2\sqrt{d-1}}$, after fixing $\alpha_{d,p}>0$ as in the cited result and taking $l\in\mathbb{N}_{>\alpha}$, there exists a solution of the semilinear equation such that

$$\phi|_{\mathcal{C}}=c_{p,d}r^{-\frac2{p-1}}+\mathcal{O}\left(r^{l-\alpha-\frac2{p-1}}\right)$$

and $\phi$ is smooth across $\mathcal{C}$. The solutions are smooth for $v<0$ and have scale-critical norm bounded below as $\tau\to0$; the source further allows finite $C^k$ regularity in place of smoothness. These conjectures describe Type II concentration around a singular self-similar exterior profile in the energy-supercritical regime. The supplied text cites constructions but explicitly presents the strengthened regularity formulation as conjectural, and gives no resolution evidence.