Counting words of minimum length in an automorphic orbit

Let u be a cyclic word in a free group Fn of finite rank n that has the minimum length over all cyclic words in its automorphic orbit, and let N(u) be the cardinality of the set and v=ϕ(u) for some ϕ∈AutFn}. In this paper, we prove that N(u) is bounded by a polynomial function with respect to |u| under the hypothesis that if two letters x,y with x≠y±1 occur in u, then the total number of occurrences of x±1 in u is not equal to the total number of occurrences of y±1 in u. A complete proof without the hypothesis would yield the polynomial time complexity of Whitehead's algorithm for Fn.