Higher-dimensional extension of the Football theorem

Let $(S^n,g_0)$ be the constant-curvature metric on $S^n$, with scalar curvature $R_0$, Ricci curvature $\operatorname{Ric}_0\cdot g_0$, and volume $V_0$. Let $(M^n,g)$ be a complete smooth Riemannian manifold of volume $V$ satisfying

$$R(g)\ge R_0,$$

and

$$\operatorname{Ric}(g)\ge \varepsilon_0\cdot \operatorname{Ric}_0\cdot g.$$

There exists a positive constant $\varepsilon_0<1$ such that

Higher-dimensional Football conjecture. For every such manifold,

$$V\le V_0.$$

The conjecture proposes extending the three-dimensional volume comparison theorem to higher dimensions; the preceding discussion notes that the three-dimensional proof uses Gauss–Bonnet and Gauss–Codazzi for isoperimetric surfaces, so a different argument would be needed in higher dimensions. Its status is not established in the supplied source.