Cluster-complex realization of positive scattering walls

Let $\varphi$ be a tropical function on the tropicalization of a cluster variety, or of a fiber of a cluster variety, let $\s{Y}$ denote that variety or fiber, and let $\f{w}$ be a scattering wall. The wall-realization conjecture. If $\varphi$ is positive at some point on $\f{w}$, then the wall-crossing formula for $\f{w}$ is the formula for some mutation in some cluster structure on $\s{Y}$; equivalently, $\f{w}$ lies in the cluster complex for some cluster structure on $\s{Y}$.

The paper states that this conjecture is the key dimension-$2$ ingredient for the preceding convexity equivalence, but gives no resolution in the supplied text beyond that setting.