Mäkelä's conjecture on ternary words with only trivial abelian squares

Let an abelian square be a word of the form $uv$ where $u$ and $v$ have the same Parikh vector. Consider infinite words over the ternary alphabet $\Sigma_3$.

Mäkelä's conjecture. There exists an infinite ternary word whose only abelian square factors are $00$, $11$, and $22$.

The survey states that this conjecture is believed to be true but remains unproved. It contrasts with the known existence of infinite ternary words having only finitely many abelian squares.