exists $\varepsilon_0 >0$ such that for all $\varepsilon$ with $0 < \varepsilon \leqslant \varepsilon_0$, the function[:]{} $$\mu
:=
\rho+\varepsilon\,v$$ is $\mathcal{ C}^\infty$ on $X_{\rm reg}$ and satisfies[:]{}

-   $\max_{ \overline{ D} } \, (\mu) < C$[;]{}

-   $X_{\rm sing} = \{ \mu = - \infty \}$[;]{}

-   $\mu$ is strongly $(n-1)$-convex in a neighborhood of $\{ \rho
    \leqslant C\}$.

Property [**(b)**]{} holds provided only that $\varepsilon < \frac{ C -
\max_{ \overline{ D}} \, (\rho) }{\max_{ \overline{ D}}\,
( v) }$. Furthermore, [**(a)**]{} is clear since $\rho$ is $\mathcal{
C}^\infty$ and since $X_{\rm sing} = \{ v = - \infty \}$. To check [**(c)**]{}, we compute Levi-forms as $(1, 1)$-forms: $$\aligned
i\,\partial\overline{\partial}\,
\widetilde{\mu}_j
&
=
i\,\partial\overline{\partial}\,
\widetilde{\rho}_j
+
\varepsilon\,
i\,\partial\overline{\partial}\,
\widetilde{v}_j
\\
&
=
i\,\partial\overline{\partial}\,
\widetilde{\rho}_j
+
\varepsilon\,
i\,\partial\overline{\partial}\,
\widetilde{u}_j
+
\varepsilon\,
i\,\partial\overline{\partial}\,
\widetilde{\sf r}_j.
\endaligned$$ Here, $\varepsilon\, i\, \partial \overline{ \partial }\, \widetilde{
u}_j$ adds positivity to $i\, \partial \overline{ \partial }\,
\widetilde{ \rho}_j$ (since $\widetilde{ u}_j$ is psh), whereas the negative contribution due to $i\, \partial \overline{ \partial}\,
\widetilde {\sf r}_j$ is bounded from below on $\{ \rho \leqslant 2C
\}$, and consequently, $\varepsilon >0$ can be chosen small enough so that $i\, \partial \overline{ \partial }\, \widetilde{ 