Determine monotonicity of the all-swaps sorting time

Determine whether the function t(n), the maximal sorting time for cyclic permutations of n labels under arbitrary swaps, is monotone in n.

Background

The function t(n) is the diameter of the graph whose vertices are cyclic permutations of n labels and whose edges correspond to swapping any two labels. The paper proves upper and lower bounds for t(n), including exact values for prime n and several lower bounds for prime-power families.

Monotonicity is unresolved because adjoining a fixed point to a permutation can decrease the associated sorting time. The authors give the example t([π_{11,3}])=9, whereas the corresponding permutation in S_12 obtained by fixing 12 has sorting time 8.