Minimal-idempotent reduction conjecture for the Kadison–Singer problem

Let $G$ be a countable discrete group, let $\beta G$ be its Stone–Čech compactification, and let $\boxed{\textstyle\text{?}}$ be a minimal idempotent, meaning an idempotent point minimal in the relevant semigroup order. Let $s_{\boxed{\textstyle\text{?}}}$ be the state associated with $\boxed{\textstyle\text{?}}$.

Minimal-idempotent reduction conjecture. If a minimal idempotent has a unique state extension, then every point has a unique state extension. Thus, for any fixed minimal idempotent $\boxed{\textstyle\text{?}}$, the Kadison–Singer conjecture is true if and only if $s_{\boxed{\textstyle\text{?}}}$ has a unique state extension.

This would reduce the global Kadison–Singer problem to checking the extension property for one minimal idempotent, but the source gives no proof or resolution.