The characterization of abelian periods for the slope $[0;\overline{2}]$

Let $\alpha=[0;\overline{2}]$, and let a Sturmian word of slope $\alpha$ be a Sturmian word whose slope is $\alpha$. Denote its abelian period set by the set of positive integers occurring as abelian periods of its factors, and write $\mathcal{Q}^+_{\alpha}$ and $\mathcal{M}_{\alpha}$ for the sets defined in the paper.

Abelian-period conjecture. The abelian period set of a Sturmian word of slope $\alpha$ is

$$\mathcal{Q}^+_{\alpha}\cup\mathcal{M}_{\alpha}.$$

The conjecture concerns the characterization of all possible abelian periods of factors of a Sturmian word for the specific slope $[0;\overline{2}]$. The surrounding examples show that the possible periods depend heavily on the arithmetic nature of the slope; the proposed description is based on computer experiments.