Conjecture on the number of inequivalent abelian squares

Two abelian squares are inequivalent when they have different Parikh vectors. Inequivalent-abelian-square bound conjecture. Every word of length $n$ contains $O(n\sqrt{n})$ inequivalent abelian squares. Computations support this asymptotic upper bound, while the paper notes that examples with order $n\sqrt{n}$ inequivalent abelian squares exist.