Classify extremal permutations with two cycles in every cyclic shift

Prove that if n>2 and a permutation π∈S_n satisfies max_{σ∈πC_n} cyc(σ)=2, then n is prime and there exists a unit a∈Z_n^× such that π(i)≡a·i (mod n) for every i.

Background

Here C_n denotes the cyclic subgroup generated by the n-cycle c=(1,2,…,n), and cyc(σ) denotes the number of cycles of a permutation σ. Lemma 3.5 shows that under the stated maximum-cycle condition, every cyclic shift σ∈πC_n must have cycle structure (n−1,1).

Conjecture 3.6 proposes a complete classification of such permutations: the order n must be prime, and π must be a multiplicative permutation modulo n. The paper states that this conjecture implies Conjecture 3.4 and that both conjectures have been verified for n≤11, leaving the general case unresolved.