Existence of nonsingular complete holomorphic foliations on Stein manifolds

Let $X$ be a Stein manifold of dimension $n\ge 2$ equipped with a Riemannian metric. A nonsingular holomorphic foliation by smooth complete closed complex hypersurfaces is a foliation of $X$ whose leaves are smooth complete closed complex hypersurfaces and which has no singularities. The foliation conjecture. Every such manifold $X$ admits a nonsingular holomorphic foliation by smooth complete closed complex hypersurfaces. In particular, every pseudoconvex domain in $\mathbb{C}^n$ admits such a foliation.

The result would strengthen the preceding construction, which produces a possibly singular foliation and does not ensure that the defining holomorphic function is noncritical. The existence of holomorphic functions without critical points is known for Stein manifolds, but obtaining one whose level hypersurfaces are all smooth, closed, and complete remains open.