Meeks–Pérez–Ros conjecture on mean curvature of foliations

Let $\mathcal{F}$ be a codimension one foliation of a complete Riemannian manifold $\overline{M}^{n+1}$. Assume that there is $K_0 \geq 0$ such that $\overline{M}$ has Ricci curvature bounded from below by $-nK_0$. Suppose that each leaf $L$ of $\mathcal{F}$ has constant mean curvature $H_L$. Meeks–Pérez–Ros conjecture. Every leaf satisfies

$$|H_L| \leq \sqrt{K_0}.$$

This conjecture gives a universal bound for the constant mean curvatures of leaves in terms of a lower Ricci-curvature bound on the ambient complete Riemannian manifold. The source attributes it to Meeks III, Pérez, and Ros; its resolution status is not established in the supplied text.