Zilber's quasiminimality conjecture for the complex exponential field

Let $\mathbb{C}_{\operatorname{exp}}$ denote the structure on the complex numbers in the language

$$\mathcal{L}=\{+,-,\cdot,1,0,\operatorname{exp}\}$$

of fields equipped with an exponential. Let $A\subseteq\mathbb{C}$ be finite. A subset of $\mathbb{C}$ is considered invariant under all automorphisms of $\mathbb{C}_{\operatorname{exp}}$ fixing $A$.

Zilber's quasiminimality conjecture. Every such invariant subset of $\mathbb{C}$ is countable or cocountable; equivalently, $\mathbb{C}_{\operatorname{exp}}$ is quasiminimal.

The conjecture predicts strong tameness and geometric properties for definable sets in powers of the complex exponential field. The source presents it as an open problem and explains that later work reduces it to a consequence of Exponential-Algebraic Closedness.