\mu > c\}$.

Let $c \in {\mathbb{R}}$ with $c < \max_M\, (\mu) < C$ be any regular value of $\mu$ and of $\mu \big \vert_M$. Let $M_{ \mu > c}'$ be any nonempty connected component of $M \cap X_{\mu > c}$. Then there is a unique connected component $Q_{ \mu > c}'$ of $X_{\mu > c} \big \backslash
M_{ \mu > c}'$ which is relatively compact in $X_{\rm reg}$ and contained in $\{ \rho < C\}$ with the property that two different domains $Q_{\mu > c}'$ and $Q_{\mu > c}''$ are either disjoint or one is contained in the other. Furthermore, for every holomorphic or meromorphic function $f$ defined in the thin tubular neighborhood $\mathcal{ V} ( M)$ of $M$, there exists a unique holomorphic or meromorphic extension $F$, constructed by means of a finite number of $(n-1)$-concave Levi-Hartogs figures and defined in $$Q_{\mu>c}'\bigcup
\mathcal{V}\big(M_{\mu>c}'\big)_{\mu>c},$$ such that $F = f$ when both functions are restricted to $\mathcal{V}\big(M_{\mu>c}'\big)_{\mu>c}$.

We only describe the modifications one must bring to the arguments of [@mp2006b].

[**1)**]{} The Levi-form of the compact $\mathcal{ C}^\infty$ boundary $\{ \mu =
c \}$ of the super-level set $\{ \mu > c\}$ (contained in $X_{\rm
reg}$) has $1$ negative eigenvalue, 