Meeks–Pérez–Ros mean-curvature bound conjecture for CMC foliations

Let $\mathfrak{F}$ be a codimension one foliation of a complete Riemannian manifold $M^{n+1}$. Assume that $M$ has absolute sectional curvature bounded from above by $1$. Suppose that each leaf $L$ of $\mathfrak{F}$ has constant mean curvature $H_L$.

Meeks–Pérez–Ros conjecture. Then

$$|H_L|\leq 1.$$

Meeks III, Pérez and Ros positively answered this conjecture when $M$ is a compact orientable $3$-manifold that is not topologically covered by $\mathbb{S}^2\times\mathbb{S}^1$, and also proved it for $M=\tilde M^3(-1)$. The general statement remains open in the context described here.