Let $(M, \omega)$ be a holomorphic symplectic manifold of (complex) dimension $2n$. Let $x$ be a point in $M$. My understanding from the discussion and answers to this MO question is that there exists a neighborhood $U \subseteq M$ of $x \in M$, and a neighborhood $V \subseteq T^* \mathbb{C}^n$, such that $V$ is symplecto-bi-holomorphic to $U$, where $T^*\mathbb{C}^n$ is given its standard homomorphic symplectic form. In other words there is a holomorphic symplectic version of Darboux's theorem. If I have miss-understood that discussion please correct me!

Now given a holomorphic function $f: \mathbb{C}^n \to \mathbb{C}$, the graph of $df$ in $T^*\mathbb{C}^n$ is a holomorphic Lagrangian submanifold.

**Question:** Given any holomorphic Lagrangian submanifold $L \subseteq M$, is it locally isomorphic to a holomorphic Lagrangian of this form?

More specifically, given any $x \in L$, do there exist a neighborhood $U \subseteq M$ of $x$ and a neighborhood $V \subseteq T^*\mathbb{C}^n$ and a symplecto-bi-holomorphism $U \cong V$, as above, such that under this isomorphism $L \cap U$ coincides with the graph of $df$ for some holomorphic function $f$ (where $f$ is defined on, say, $V \cap \mathbb{C}^n$)?

Remark: We know (see here) that there is no holomorphic version of the Weinstein Lagrangian Neighborhood theorem. This question is morally asking if there is a much weaker local version that still holds holomorphically.