Existence of supramenable groups without the fixed-point property for cones

A supramenable group is a group $G$ such that every $G$-set $X$ supports an invariant finitely additive measure normalized on any given non-empty subset $A\subseteq X$. The fixed-point property for cones is the property characterized in the source by the existence of a $G$-invariant conditional mean on $\ell^\infty(G)$. Supramenability versus the fixed-point property. There exist supramenable groups $G$ without the fixed-point property for cones. The fixed-point property for cones implies supramenability, and the conjecture asks whether this implication is strict. The source provides no resolution.