Conjectured conormal regularity for Type I singularity formation

Let $\mathcal{X}=(\{v\leq 0\}\cup \{t\leq -0.9\})\cap\{-1\leq u<0\}$, and let $\phi$ be a solution arising from smooth initial data in the stated Type I self-similar stability results, with singularity formation renormalised to $t=0$. Write $\phi_0$ for the corresponding self-similar solution, $\mathcal{C}$ for the backward light cone, and let $\{u\partial_u,x_i\partial_{x_j}-x_j\partial_{x_i},r\partial_t,1\}^\alpha$ denote products of the listed vector fields of order $|\alpha|$. Type I regularity conjecture. There exists $\epsilon>0$, depending on the equation, such that for every $k\in\mathbb{N}$ and every $0\leq|\alpha|\leq k$, the following estimates hold on $\mathcal{C}$:

$$\left|\{u\partial_u,x_i\partial_{x_j}-x_j\partial_{x_i},r\partial_t,1\}^\alpha(\phi-\phi_0)\right|\lesssim_{|\alpha|}r^\epsilon$$

for the wave-map equation with $d\geq3$, and

$$\left|\{u\partial_u,x_i\partial_{x_j}-x_j\partial_{x_i},r\partial_t,1\}^\alpha(\phi-\phi_0)\right|\lesssim_{|\alpha|}r^{\epsilon-1}$$

for the equation with $d=7$ and $p=3$. These estimates assert the additional conormal and $r\partial_t$ regularity needed for the exterior stability argument. The conjecture concerns precise quantitative control not supplied by the preceding stability theorem, and its resolution is not indicated here.