Uniform a priori estimate conjecture for the boundary conformal equation

Let $(M,g)$ be a compact $n$-dimensional Riemannian manifold with boundary $\partial M$. Suppose that $M$ is of positive type and is not conformally equivalent to $\mathbb S^n_+$. Assume further that $n=3$ or that $M$ is locally conformally flat with $\partial M$ umbilic and $n\geq 3$. For a small $\delta>0$, let $C(M,g,\delta)>0$. For any solution $u>0$ of the referenced boundary equation with $0\leq \kappa\leq 1-\delta$, assume $0<\alpha<1$.

Uniform a priori estimate conjecture. There exists $C(M,g,\delta)>0$ such that

$$C^{-1}\leq u\leq C\qquad\text{and}\qquad \|u\|_{C^{2,\alpha}(M)}\leq C.$$

The proposed estimate would provide uniform compactness away from the endpoint $\kappa=1$ and is intended to support a topological-degree proof of the existence conjecture. The statement is presented as a conjecture, and no resolution is supplied in the source.