Kucherov–Rytter–Rytter–Wendlandt conjecture on non-equivalent abelian squares

For a binary word of length $n$, let $XNL(n)$ denote the maximum number of non-equivalent abelian squares it contains. Kucherov et al.'s conjecture.

$$XNL(n)=O(n^{1.5}).$$

The conjecture proposes a polynomial upper bound matching the scale suggested by known lower-bound constructions, but the source gives no resolution.