Binary geometry existence conjecture for polygon face complexes

Let $\Delta$ be the face complex of an $n$-gon. A binary geometry for $\Delta$ is a binary geometry whose underlying flag complex is $\Delta$. Binary geometry existence conjecture. Such a binary geometry exists if and only if

$$n\in\{4,5,6,8\}.$$

This conjecture classifies the polygonal face complexes admitting binary geometries; the supplied text gives examples for the square, pentagon, hexagon, and octagon but does not state whether the classification is resolved.