  The purpose of this article is study rigidity of free boundary minimal
two-disks that locally maximize the modified Hawking mass on a Riemannian
three-manifold with positive lower bound on its scalar curvature and mean
convex boundary. Assuming the strict stability of {\Sigma}, we prove that a
neighborhood of it in M is isometric to one of the half de Sitter-Schwarzschild
space.
