Boundary area rigidity conjecture under positive Ricci curvature

Let $M^n$ be a compact Riemannian manifold with boundary $\Sigma=\partial M$, $Ric\geq n-1$, and second fundamental form $\Pi\geq 0$. Boundary area rigidity conjecture.

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

Moreover, equality should imply that $(M^n,g)$ is isometric to the hemisphere

$$\mathbb{S}^n_+=\\{x\in\mathbb{R}^{n+1}:|x|=1,\\ x_{n+1}\geq 0\\}\subset\mathbb{R}^{n+1}.$$

The statement is proposed as a sharp area bound with rigidity in the positive-Ricci setting; the supplied text gives no resolution.