Conjecture on equality of support, quantum orbit and minimal invariant subset

Let $\mathcal{G}$ be a compact quantum group acting on a compact Hausdorff space $X$. For $x\theta X$, let $rm Orb_x$ denote the orbit of $x$, let $rm supp\,\mu_x$ denote the support of the associated measure $mu_x$, and let $rm \mathcal{M}_x$ denote the minimal invariant subset associated with $x$. Support–orbit conjecture. For every $x\theta X$, one has

$$\operatorname{supp}\mu_x=\mathcal{M}_x=\operatorname{Orb}_x.$$

The preceding theorem proves $\operatorname{supp}\mu_x\subseteq\mathcal{M}_x\subseteq\operatorname{Orb}_x$, with equality between the first two sets when the Haar measure is faithful. Equality is also known in the coamenable and countable cases cited by the source; the general assertion remains open.