=
i\,\partial\overline{\partial}\,
\widetilde{\rho}_{j(p_a)}
+
\varepsilon\,i\,\partial\overline{\partial}\,
\widetilde{u}_{j(p_a)}
+
\varepsilon\,i\,\partial\overline{\partial}\,
\widetilde{\sf r}_{j(p_a)}.$$ We choose $\varepsilon >0$ so small that the remainder $\varepsilon
\,i \, \partial \overline{ \partial}\, \widetilde{\sf r}_{ j(p_a)}$ does not perturb positivity on $\widetilde{ \sf C}_{ j ( p_a)}$ for every $a = 1, \dots A$, and we get that $i\, \partial \overline{
\partial}\, \widetilde{\mu }_{ j(p_a)}$ is still positive on $\widetilde{ F}_{ \widetilde{ q}}$ for every $\widetilde{ q} \in
\widetilde{ \sf C}_{ j ( p_a )}$, and every $a = 1, \dots A$.

Let $q \in \{ \rho = C\} \cap 
X_{\rm reg}$. Then $q \in V_{ j ( p_a)}$ for some $a$. We set $\widetilde{ q} := \varphi_{ j( p_a)} ( q)\in \widetilde{ \sf
C}_{ j ( p_a)}$ and we define: $$F_q
:=
\big(
d\,\varphi_{j(p_a)}
\big)^{-1}
\Big(
\widetilde{F}_{\widetilde{q}}\cap
T_{\widetilde{q}}\,{\sf A}_{j(p_a)}
\Big).$$ Then the complex linear spaces $\widetilde{
F}_{ \widetilde{ q}}$ and $F_q$ are at least of dimension $2$ and the Levi-form of $\mu$ is positive on any $1$-dimensional subspace $E_q \subset F_q \cap T_q^c \, \Lambda$.

Next, applying Morse transversality theory, we may perturb $\mu$ in $X_{\rm reg} \cap \{ \rho < 2\,C\}$ in an arbitrarily small way, so that[^6]:

-   [*$\mu$ is a Morse function on $X_{\rm 