The classification conjecture for unsplittable cyclic $(n_3)$ configurations

A cyclic $(n_3)$ configuration is a configuration whose Levi graph is a cyclic Haar graph; write $H(n,S)$ for the Haar graph associated with the integer $n$ and connection set $S$. The configuration is unsplittable when it is neither point-splittable nor line-splittable.

Classification conjecture. A cyclic $(n_3)$ configuration is unsplittable if and only if its Levi graph belongs to one of the following three infinite families:

1. $H(n, \{0,1,3\})$ for $n\geq 7$;
2. $H(3n, \{0,1,n\})$ for $n\geq 2$;
3. $H(3n, \{0,1,n+1\})$ for $n\geq 4$ where $n\not\equiv 0\pmod 3$.

The conjecture is motivated by the known results on splittability of cyclic Haar graphs and by experimental evidence, but the classification of all remaining cyclic $(n_3)$ configurations is left open.