CoNP-completeness conjecture for exact-four-occurrence 3-Horn connectivity

Prove that the Boolean connectivity problem for 3-Horn formulas with exactly three literals per clause and exactly four occurrences of every variable is coNP-complete.

Background

The authors explicitly formulate the expected classification of the unresolved exact-occurrence case as a conjecture. Their conjecture is based on the correspondence between the known bounded-occurrence results for Boolean connectivity of 3-Horn formulas and the corresponding results for Monotone NAE 3-SAT.

This is the same unresolved case identified immediately preceding the conjecture; it is retained here as the paper’s explicit conjectural formulation of that open problem.

References

We conjecture that {\sc Conn E3-Horn-E4} is $\mathsf{coNP}$-complete, since {\sc Monotone NAE 3-SAT} remains $\mathsf{NP}$-complete even when each clause has exactly three literals and each variable appears exactly four times, and because the currently known complexity results for {\sc Conn $3$-Horn} with bounded variable occurrences mirror those for {\sc Monotone NAE 3-SAT} under the same restrictions.

— The Complexity of Boolean Connectivity Problem of $k$-Horn Formulas  (2608.19569 - Horiyama et al., 20 Aug 2026) in Section Conclusion