Fazekas–Mammoliti–Mercaş–Simpson extension of the Fici–Saarela conjecture

Let $w$ be a binary word of length $n$. An abelian square is a factor consisting of two consecutive factors with the same Parikh vector. An abelian square of the form $a^{2i}$ or $b^{2i}$, for a positive integer $i$, is called trivial.

Fazekas–Mammoliti–Mercaş–Simpson conjecture. A binary word of length $n$ contains at least

$$\left\lfloor \frac{n}{4}\right\rfloor$$

abelian squares, and if it contains exactly $\left\lfloor \frac{n}{4}\right\rfloor$ abelian squares, then all the abelian squares are trivial.

This is presented as an extension of the Fici–Saarela conjecture. The supplied text gives no resolution of either the lower bound in full generality or the equality characterization.