Positivity conjecture for the free boundary Brown–York mass

Let $(M,g)$ be a compact Riemannian three-manifold with boundary $\partial M=\Sigma\cup S$, where $S$ and $\Sigma$ meet orthogonally, $\Sigma$ has positive Gauss curvature, and its boundary geodesic curvature satisfies $k_{h_\Sigma}=1$. Define the free boundary Brown–York mass by

$$m^{FB}_{BY}(M):=\int_{\Sigma}(H_e-H)\,\operatorname{d}vol_{h_\Sigma}.$$

Assume that $\Sigma$ is strictly mean convex, that $(M,g)$ satisfies $\operatorname{Sc}\geq 0$, and that $H^S\geq 2$. Free boundary Brown–York positivity conjecture. Then

$$m^{FB}_{BY}(M)\geq 0.$$

Equality holds if and only if $M$ is the Euclidean domain bounded by a part of the unit sphere and a convex free boundary surface. This conjecture formulates positivity and rigidity for the newly defined free boundary Brown–York mass; the source presents it as open.