Unsplittability conjecture for cyclic configurations of rank at least four

A cyclic $(n_k)$ configuration is a cyclic configuration in which every point lies on $k$ lines and every line contains $k$ points. Unsplittability conjecture. All cyclic $(n_k)$ configurations for $k \geq 4$ are unsplittable.

The claim is motivated by the fact that no splittable cyclic $(n_k)$ configuration with $k \geq 4$ was obtained, despite constructions of geometric point- and line-splittable configurations for every $k$.