Boundary area conjecture under nonnegative Ricci curvature and convexity

Let $M^n$ be a compact Riemannian manifold with $Ric\geq 0$ and second fundamental form $\Pi\geq 1$ on $\partial M$. Boundary area conjecture.

$$|\partial M|\leq |\mathbb{S}^{n-1}|.$$

The text presents this as a consequence of the preceding positive-solution conjecture, and the supplied status evidence identifies the associated sharp spherical inequality as proved by Beckner; however, no direct resolution of this geometric conjecture is supplied here.