Binary-alphabet extremal conjecture for abelian-square factors

For a finite word, an abelian-square factor is a factor formed by concatenating two anagrams of one another. Let $n$ be a positive integer, and let $k$ be the number of distinct abelian-square factors in a word of length $n$. Binary-alphabet extremal conjecture. If a word of length $n$ contains $k$ distinct abelian-square factors, then there exists a binary word of length $n$ containing at least $k$ distinct abelian-square factors. This conjecture asserts that binary words attain the maximum possible number of distinct abelian-square factors among words of each fixed length; the source leaves it as an open question.