Wilkie's local definability conjecture for holomorphic functions

Let $\mathcal{A}$ be a family of complex holomorphic functions. Let $z\in\mathbb{C}^{n}$, let $U$ be an open neighbourhood of $z$, and let $f:U\rightarrow\mathbb{C}$ be holomorphic and locally $\emptyset$-definable in $\mathbb{R}_{\mathcal{A\upharpoonright}}$. Let $\mathcal{P}_{G}$ denote the family of polynomials with Gaussian rational coefficients. Wilkie's conjecture. There is an open box with Gaussian rational corners $\Delta\subseteq U$ such that $z\in\Delta$ and $f\upharpoonright\Delta$ can be obtained from $\mathcal{A}\cup\mathcal{P}_{G}$ by finitely many applications of composition, Schwarz reflection, taking partial derivatives, and extracting implicit functions.

The conjecture proposes a uniform complex-analytic description of locally definable holomorphic functions, extending Wilkie's generic-point result to every point. The paper's introduction states that this description is not complete around non-generic points, so the conjecture is refuted by the results of the paper.