Schoen's curvature estimate conjecture for stable minimal hypersurfaces

Let $M^4$ be a Riemannian manifold, let $B_{r_0}=B_{r_0}(x)\subset M^4$, and let $\Sigma^3\subset B_{r_0}$ be an immersed $2$-sided stable minimal hypersurface with $\partial\Sigma\subset\partial B_{r_0}$. Suppose that $|K_M|\leq k^2$ and $r_0<\rho_1(\pi/k,k)$. Schoen's curvature estimate conjecture. There is a constant $C=C(k)$ such that, for every $0<\sigma\leq r_0$,

$$\sup_{B_{r_0-\sigma}}|A|^2\leq C\,\sigma^{-2}.$$

The conjecture proposes a local curvature estimate under bounded ambient sectional curvature and a radius restriction; the supplied text gives no resolution evidence.