Several Tokens in Herman's algorithm

We have a cycle of N nodes and there is a token on an odd number of nodes. At each step, each token independently moves to its clockwise neighbor or stays at its position with probability 2 . If two tokens arrive to the same node, then we remove both of them. The process ends when only one token remains. The question is that for a fixed N , which is the initial configuration that maximizes the expected number of steps E(T ). The Herman Protocol Conjecture says that the 3-token configuration with distances b N3 c and d N 3 e maximizes E(T ). We present a proof of this conjecture not only for E(T ) but also for E ( f(T ) ) for some function f : N → R+ which method applies for different generalizations of the problem.