Desarguesian conjecture for cyclic projective planes

A cyclic projective plane is a projective plane admitting a regular action by a cyclic group. A projective plane is Desarguesian when it satisfies Desargues's theorem, and a perfect difference set gives rise to a cyclic projective plane. Desargues's conjecture. Every finite cyclic projective plane, and hence every corresponding perfect difference set, is Desarguesian.

This conjecture would provide an alternative route to the paper's main theorem because Desargues's theorem imposes strong geometric restrictions. The source explicitly says that the conjecture is open; Hall described the assertion as highly plausible.