Immersed version of the uniform curvature estimate for stable free boundary minimal surfaces

Let $(M^3,g)$ be a compact Riemannian $3$-manifold with nonempty boundary, and let $(\Sigma,\partial\Sigma)\subset(M,\partial M)$ be a compact, properly immersed stable minimal surface with free boundary. The surface need not be embedded.

Immersed curvature estimate conjecture. The uniform curvature estimate

$$\sup_{x\in\Sigma}|A|^2(x)\leq C_2$$

holds for a constant $C_2>0$ depending only on the geometry of $M$ and $\partial M$.

The paper proves this estimate for compact, properly embedded stable minimal surfaces with free boundary. The conjecture asks whether embeddedness can be replaced by immersion, in analogy with interior curvature estimates for stable immersed minimal surfaces in $3$-manifolds.