The magnetic gap-labelling conjecture for strongly minimal actions

Let $\Sigma$ be a Cantor set equipped with an action of ${\mathbb Z}^p$, and suppose that the action is strongly minimal. Let $\mu$ be a preserved Borel probability measure, let $\Theta$ be the skew-symmetric matrix determining the magnetic multiplier $\sigma$, and let $\mathbb Z_I[\mu]$ and $\operatorname{Pf}(\Theta_I)$ be as in the magnetic gap-labelling conjecture for minimal actions.

Magnetic gap-labelling conjecture for strongly minimal actions. If the action is strongly minimal, then the containment in the magnetic gap-labelling conjecture for minimal actions can be replaced by equality.

This conjecture identifies strong minimality as a condition that may ensure the full explicit gap-labelling group, rather than merely an upper bound. The source proves the assertion for strongly minimal actions of ${\mathbb Z}^3$, while its validity in higher dimensions is not clear.