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≠4m≠4 can be cyclically ordered such that the difference of every pair of neighbouring elements is coprime to the module v≔m2+m+1v≔m2+m+1. We prove that this is the case whenever the number ω(v)ω(v) of different prime divisors of vv 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.