Binary Words Containing Few Abelian Squares

Fici and Saarela ([2]) conjectured that a binary word of length n contains at least $\lfloor n\mathrm{/}4\rfloor$ abelian squares. We slightly extend this conjecture and show that it holds in some special cases. In all other cases we have the following: given a Parikh vector over a two letter alphabet we produce a word with that Parikh vector which we conjecture contains the least possible number of abelian squares.