Fici–Saarela minimum distinct abelian-square conjecture

For a binary linear word of length $n$, let $MDL(n)$ be the minimum number of distinct abelian squares it contains. Fici–Saarela's conjecture.

$$MDL(n)=\lfloor n/4\rfloor,$$

and the only words of length $4k+3$ containing exactly $k$ distinct abelian squares are $a^{2k+1}ba^{2k+1}$ and its complement. The conjecture gives both the sharp minimum and the stated equality cases for distinct abelian squares in binary words.