Vanishing-order conjecture for the umbilicity tensor and Weyl curvature

Let $g$) be a smooth Riemannian metric defined on the unit half $n$-ball $B^+_1$, where $4\leq n\leq24$. Suppose there is a sequence of positive solutions $\{u_i\}$ of

$$\begin{cases} L_g u_i+K u_i^{p_i}=0 & \text{in } B^+_1,\\ B_g u_i=0 & \text{on } \overline{B^+_1}\cap\mathbb{R}^{n-1}, \end{cases}$$

with $p_i\in(1,\frac{n+2}{n-2}]$, such that for every $\varepsilon>0$ there is a constant $C(\varepsilon)>0$ satisfying $\sup_{B^+_1\setminus B^+_\varepsilon}u_i\leq C(\varepsilon)$ and $\lim_{i\to\infty}\sup_{B^+_1}u_i=\infty$. Vanishing-order conjecture. The umbilicity tensor $T_g$ satisfies

$$|T_g|(x)\leq C|x|^m,\qquad x\in\overline{B^+_1}\cap\mathbb{R}^{n-1},$$

for some integer $m>\frac{n-4}{2}$. Moreover, if $n\geq6$, then

$$|W_g|(x)\leq C|x|^\ell,\qquad x\in B^+_1,$$

for some integer $\ell>\frac{n-6}{2}$. The asserted estimates describe the high-order vanishing of conformally invariant curvature quantities at a boundary blow-up point; establishing them would constrain blow-up for the Yamabe problem with boundary in dimensions $4\leq n\leq24$.