Hunter Snevily (snevily@uidaho.edu) wrote back to me:

Hi- Jerry Griggs found a counterexample 2,2,2,-1,-1,-1,-1,-1,-1 I have not checked this out but Jerry is reliable. sorry hs

Also, David Callan (callan@stat.wisc.edu) wrote:

The set {1,1,1,-1,-1,-1} has {r1,r2,...,rn} = {3, 12, 10, 12, 3, 1}, not unimodal. DC

Jim Propp