Extremal bound conjecture for inequivalent abelian squares

Two abelian squares are inequivalent when they have different Parikh vectors, where the Parikh vector records the number of occurrences of each letter. Let $n$ be a positive integer and consider words of length $n$. Inequivalent abelian-square bound conjecture. Every word of length $n$ contains at most $\Theta(n\sqrt{n})$ inequivalent abelian squares. The source notes that computations support this conjecture and that the matching lower-order magnitude $\Omega(n\sqrt{n})$ is attainable, while the asserted universal upper bound remains open.