The Hesse configuration conjecture for complex Steiner triple systems

A Steiner triple system is a configuration of points and lines in which every line passes through exactly three points. Consider complex Sylvester–Gallai configurations embedded in $\mathbb{C}^2$ whose lines all pass through exactly three points.

Hesse configuration conjecture. Aside from the Hesse configuration, there are no complex Sylvester–Gallai configurations where each line passes through exactly three points.

Equivalently, the conjecture says that the only Steiner triple system embeddable in $\mathbb{C}^2$ is the Hesse configuration. The claim concerns the classification of complex realizations of Steiner triple systems and remains open in the source.