Wandering-point unique extension conjecture

Let $G$ be a countable discrete group acting on its Stone–Čech remainder $G^*$. A point $\boxed{\textstyle\text{?}}\in G^*$ is wandering if it has an open neighborhood whose translates by distinct group elements are pairwise disjoint. Let the state corresponding to evaluation at $\boxed{\textstyle\text{?}}$ be the associated state on the diagonal algebra.

Wandering-point conjecture. If $\boxed{\textstyle\text{?}}$ is wandering for the $G$-action on $G^*$, then the state corresponding to evaluation at $\boxed{\textstyle\text{?}}$ extends uniquely to a state on $B(\ell^2(G))$.

The result cited in the source establishes unique extension for states arising from rare ultrafilters, and the conjecture asks whether the broader wandering-point condition suffices.