The higher-dimensional volume comparison conjecture

Let $(S^n,g_0)$ be the sphere with its constant-curvature metric, scalar curvature $R_0$, Ricci curvature represented by $Ric_0\cdot g_0$, and volume $V_0$. For each $n\geq 3$, let $(M^n,g)$ be a complete smooth Riemannian manifold of volume $V$.

Higher-dimensional volume comparison conjecture. There exists a positive $\epsilon_0(n)<1$ such that, whenever

$$R(g)\geq R_0,$$ $$Ric(g)\geq\epsilon_0(n)\cdot Ric_0\cdot g,$$

then

$$V\leq V_0.$$

The claim generalizes the preceding volume comparison theorem from dimension three to all dimensions $n\geq3$; the source gives no resolution.