Non-recurrent-point range conjecture for the completely positive map

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. Let $\boxed{\textstyle\text{?}}_{\boxed{\textstyle\text{?}}}$ be the completely positive map associated with $\boxed{\textstyle\text{?}}$, and let $\boxed{\textstyle\text{?}}(\boxed{\textstyle\text{?}}_{\boxed{\textstyle\text{?}}})$ denote its range.

Non-recurrent-point range conjecture. If $\boxed{\textstyle\text{?}}$ is non-recurrent, then

$$\mathcal{R}(\psi_{\boxed{\textstyle\text{?}}})=B(\ell^2(G)).$$

This is a stronger operator-algebraic formulation associated with the preceding unique-extension conjecture, but the source supplies no resolution.