The lower mean curvature bound in Gromov's mean-of-the-mean-curvature conjecture

Gromov conjectured that for a compact Riemannian manifold $X$ with boundary, the total mean curvature $\int_{\partial X} H$ is bounded above by a constant depending only on the intrinsic geometry of $\partial X$ and a lower bound on the scalar curvature of $X$. Previous results towards this conjectu