$$X_{\rm sing}
=
\big\{
v=-\infty
\big\}.$$ As in Section 2, if ${\sf A}_j = \varphi_j (U_j)$ is represented in a local ball $\widetilde{ \sf B}_j \subset {\mathbb{C}}^{ N_j}$ of radius $r_j>0$ centered at $z_j \in {\mathbb{C}}^{ N_j}$ as the zero-set $\{ g_{j,\nu} = 0\}$ of finitely many functions $g_{ j,\nu}$ holomorphic in a neighborhood of the closure of $\widetilde{ \sf B}_j$, the local ambient $\widetilde{ v}_j : \widetilde{ \sf B}_j \to \{ - \infty \} \cup {\mathbb{R}}$ is essentially of the form[^5]: $$\widetilde{ v}_j
=
{\rm log}\, 
\Big(
\sum_\nu\,\vert 
g_{j,\nu} 
\vert^2 
\Big)
-
\frac{1}{r_j^2-\vert z-z_j\vert^2}.$$ Thus, locally on each $\widetilde{ \sf B}_j$, the function $v$ we pick from [@de1990] is of the form: $$\widetilde{v}_j
=
\widetilde{u}_j
+
\widetilde{\sf r}_j,$$ with $\widetilde{ u}_j$ strictly psh, $\mathcal{ C}^\infty$ on $\widetilde{ \sf B}_j \big \backslash \big[ {\sf A}_j \big]_{\rm
sing}$, equal to $\{ - \infty \}$ on $\big[ {\sf A}_j \big]_{\rm
sing}$ and with a remainder $\widetilde {\sf r}_j$ which is $\mathcal{
C}^\infty$ on the whole of $\widetilde{ \sf B}_j$. Notice that each $\widetilde{ v}_j$ is $L_{\rm loc}^\infty$.

3.4. Modified strongly $(n-1)$-convex exhaustion function $\mu$ {#modified-strongly-n-1-convex-exhaustion-function-mu .unnumbered}
---------------------------------------------------------------

Pick a constant $C > 0$ such that $\max_{ \overline{ D}}\, (\rho) < C$.

There 