My question is of local nature.

Let $$f:\mathbb C^n\to\mathbb R$$ be a $C^\infty$ function that vanishes at $0\in \mathbb C^n$, with non-zero derivative.

Then, around $0\in \mathbb C^n$, $$M:=f^{-1}(0)$$ is a CR manifold. Let me assume that $M$ is the simplest possible kind of CR manifold, namely that it is foliated by real-codimension-one complex submanifolds.

[Equivalently, for those who don't know what CR manifolds are, consider the hyperplane distribution $L:=TM\cap i\cdot TM\subset TM$. I require the distribution $L$ to be integrable, i.e., to come from a (real codimension $1$) foliation of $M$.]

Under the above assumptions, is $f^{-1}\big([0,\infty)\big)$ locally isomorphic to $$\big\{(z_1,...z_n)\in\mathbb C^n\,:\,\mathrm{im}(z_1)\ge 0\big\}?$$

I.e., does there exist a neighbourhood $U\subset f^{-1}([0,\infty))$ of zero and an isomorphism $\varphi:U\to \big\{z\in\mathbb C^n\,:\,\sum|z_i|^2<1,\,\mathrm{im}(z_1)\ge 0\big\}$ which is holomorphic in the interior and smooth all the way to the boundary.