Disks area-minimizing in mean convex Riemannian $n$-manifolds

We prove the validity of an inequality involving a mean of the area and the length of the boundary of immersed disks whose boundaries are homotopically non-trivial curves in an oriented compact manifold which possesses convex mean curvature boundary, positive escalar curvature and admits a map to ${\mathbb{D}}^2\times T^n$ with nonzero degree, where ${\mathbb{D}}^2$ is a disk and $T^n$ is an $n$-dimensional torus. We also prove a rigidity result for the equality case when the boundary is totally geodesic. This can be viewed as a partial generalization of a result due to Lucas Ambr'ozio in \cite{AMB} to higher dimensions.