An abelian periodicity lemma

We write x≺y when x and y are vectors with each element of x less than or equal to the corresponding element of y and P(w) for the Parikh vector of a word w. A word w has abelian period p if it has the form u0u1⋯umum+1 with |u0|≤p, |ui|=p for i=1,…m and |um+1|≤p, and P(u0)≺P, P(u0)=P for i=1,…,m and P(um+1)≺P for some vector P. Blanchet-Sadri et al. conjectured that if a word has abelian periods pd and qd, where gcd⁡(p,q)=1, and length at least 2pqd−1 then the number of distinct letters appearing in the word is at most d, and that under certain conditions the bound may be reduced to 2pqd−2. We prove their conjecture.