Filling-radius comparison with the round sphere

Let $(M^n,g)$ be a complete Riemannian manifold, and let $(S^n,g_0)$ be a round sphere. Choose the radius of the round sphere so that the filling radius of $(M^n,g)$ equals that of $(S^n,g_0)$. Filling-radius sphere comparison conjecture.

$$V_{(M^n,g)}(R) \ge V_{(S^n,g_0)}(R)$$

for all $R$. This is proposed as a sharp comparison for ball volumes at fixed filling radius; the supplied text gives no resolution status.