Ricci lower-bound conjecture for mean curvature of CMC foliations

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

Ricci lower-bound conjecture. Then

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

This is presented as a generalization of the earlier mean-curvature bound conjecture. The paper establishes related bounds under additional hypotheses, including a sign condition on the mean-curvature function, but the full assertion remains open.