Fici–Saari conjecture on the minimum number of binary abelian square factors

Let $f_2(n)$ be the least number of distinct abelian square factors in a binary word of length $n$. An abelian square is a factor whose two halves are abelian equivalent, that is, have the same Parikh vector.

Fici–Saari conjecture. Every binary word of length $n$ contains at least $\floor{n/4}$ distinct abelian square factors; equivalently,

$$f_2(n)=\floor{n/4}.$$

The conjecture is presented as being supported by computer experiments. Earlier results show that $f_2(n)$ is unbounded, but the asserted exact minimum remains open.