Coprime ordering of cyclic planar difference sets

Motivated by a question about uniform dessins d’enfants, it is conjectured that every cyclic planar difference set of prime power order m ≠ 4 can be cyclically ordered such that the difference of every pair of neighbouring elements is coprime to the module v ≔ m 2 + m + 1 . We prove that this is the case whenever the number ω ( v ) of different prime divisors of v is less than or equal to 3. To achieve this we consider a graph related to the difference set and show that it is Hamiltonian.