On hyperbolicity and tautness modulo an analytic subset of Hartogs domains

Let $X$ be a complex space and $H$ a positive homogeneous plurisubharmonic function $H$ on $X\times {\text{C}}^m$. Consider the Hartogs-type domain ${\mathrm{\Omega }}_H(X):=\{(z,w)\in X\times {\text{C}}^m:H(z,w)<1\}$. Let $S$ be an analytic subset of $X$. We give necessary and sufficient conditions for hyperbolicity and tautness modulo $S\times {\text{C}}^m$ of ${\mathrm{\Omega }}_H(X)$, with the obvious corollaries for the special case of Hartogs domains.