Characterize when the all-swaps circular sorting bound is tight

Determine whether the maximal sorting time t(n) for cyclic permutations under arbitrary swaps satisfies t(n)=n−2 if and only if n is prime.

Background

The paper defines t(n) as the maximum, over cyclic permutations of n labels, of the minimum number of arbitrary swaps needed to sort the cyclic permutation to the trivial one. Observation 3.1 establishes the universal upper bound t(n)≤n−2, while Corollary 3.3 proves equality for every prime n.

The conjecture asserts that primality is also necessary for equality. The authors report that the related computational evidence, and the conjectures in this section, have been verified for n≤11, but they do not establish the characterization for all composite integers.