Fraenkel–Paterson–Simpson minimum non-equivalent abelian-square conjecture

For a binary linear word of length $n$, let $MNL(n)$ be the minimum number of non-equivalent abelian squares it contains. Fraenkel–Paterson–Simpson's conjecture. For every positive integer $n$,

$$MNL(n)=\lfloor n/4\rfloor.$$

When $n=4k+3$, the bound is attained only by $a^{2k+1}ba^{2k+1}$ and $(ab)^{2k+1}a$, together with their complements; when $n$ is not congruent to $3$ modulo $4$, the extremal words are obtained by removing $1$, $2$, or $3$ letters from an end. This conjecture concerns the sharp minimum and the equality cases for non-equivalent abelian squares in binary words.