Gromov's upper bound conjecture for total mean curvature of spin fill-ins

Let $M$ be a compact Riemannian manifold without boundary which bounds a compact manifold, and let $\sigma\in\mathbb{R}$. For a compact Riemannian manifold $X$ with boundary $\partial X=M$, let $H$ denote the unnormalized mean curvature of the boundary and suppose that

$$\operatorname{scal}_X\ge \sigma.$$

Gromov's upper bound conjecture. There exists a constant $C(M,\sigma)$ such that for every such $X$,

$$\int_M H\le C(M,\sigma).$$

The conjecture asserts that the total mean curvature of the boundary cannot be made arbitrarily large when the boundary metric and a lower scalar-curvature bound for the fill-in are fixed. The paper proves this assertion for spin manifolds, with a constant that also depends on a lower bound for the boundary mean curvature; the unrestricted statement is the conjectural claim being addressed.