Thurston's CMC foliation conjecture for almost Fuchsian manifolds

An almost Fuchsian manifold is a quasi-Fuchsian hyperbolic three-manifold containing a closed minimal surface homeomorphic to a fixed closed oriented surface of genus at least two, with all principal curvatures between $-1$ and $1$. A constant mean curvature (CMC) foliation is a foliation by closed oriented incompressible surfaces whose mean curvature is constant on each leaf. Thurston's CMC foliation conjecture. Any almost Fuchsian manifold is globally, monotonically, uniquely foliated by closed oriented incompressible surfaces of constant mean curvature. The conjecture concerns the existence and uniqueness of a global CMC foliation extending through the entire almost Fuchsian manifold; the supplied text does not state whether it has been resolved.