Conjecture that binary words maximize the number of abelian square factors

For a finite word, an abelian square is a factor formed by concatenating two anagrammatic words. Binary-word maximality conjecture. If a word of length $n$ contains $k$ distinct abelian square factors, then some binary word of length $n$ contains at least $k$ distinct abelian square factors. The paper leaves this as an open question after proving quadratic abelian-square richness for several binary-word classes; it is presented as a conjecture supported by the authors' investigation.