Non-recurrent-point unique extension conjecture

Let $G$ be a countable group with no finite subgroups, let $\beta G$ be its Stone–Čech compactification, and let $\boxed{\textstyle\text{?}}\in\beta G$ be non-recurrent, meaning that it is not recurrent for the induced $G$-action. Let $s_{\boxed{\textstyle\text{?}}}$ be the state associated with $\boxed{\textstyle\text{?}}$.

Non-recurrent-point conjecture. If $\boxed{\textstyle\text{?}}$ is non-recurrent, then $s_{\boxed{\textstyle\text{?}}}$ extends uniquely to $B(\ell^2(G))$.

The preceding result in the paper shows that non-recurrent points satisfy a necessary injective-envelope condition for unique extension, motivating this conjecture; its resolution is not supplied here.