Possibility of 22 < -3/2 for three-element permutations with m > 3

Determine whether the swap-distance optimality score 22, defined as 22 = ((d)r - (d))/((d)r - (d)min) where (d) is the average swap distance for a given arrangement, (d)r is the expected value under a random permutation of the same probabilities, and (d)min is the minimum over all permutations, can take values strictly smaller than -3/2 in the case n = 3 (three constituents) when the number of non-zero probability orders satisfies m > 3.

Background

Appendix B develops bounds on the normalized optimality score 22 for the average swap distance. For m = 2, the authors derive a closed-form expression for 22min(n) and show it increases with n, with 22min(3) = -3/2. For n = 3 and m = 3, they show that 22min(3) > -3/2 + ε (approaching -3/2 from above).

They further note that values arbitrarily close to -3/2 + ε can be constructed for any m by concentrating most probability mass on two orders at maximum swap distance and distributing a small residue across the remaining orders. However, it remains unresolved whether 22 can be strictly smaller than -3/2 when n = 3 and m > 3.