On the Relation between a Taut Manifold $X$ Modulo $Delta_X$ and Hyperbolic Modulo $Delta_X$ Which Is Contained in a Hypersurface of $X$

Let $X$ be a n-dimensional Stein (connected complex) manifold or a compact one whose universal covering is a domain in $mathbf{C}^n$ or a Stein manifold. Let $Delta_X$ be the degeneration locus of Kobayashi pseudodistance of $X$ which is contained in a hypersurface $S$ of $X$. Then $X$ is hyperbolic modulo $S$ and taut modulo $S$.