The simplex-like compactification conjecture for cluster varieties

Let $Y$ be a cluster variety that is locally acyclic and of full rank. Write $Y_{>0}$ for its positive part and $\Omega(Y_{>0})$ for its natural top form. The simplex-like compactification conjecture. There is a compactification $X$ of $Y$ such that $(X,X_{\geq 0})$ is a simplex-like positive geometry satisfying

$$X_{>0}=Y_{>0},\qquad Y=X\setminus\partial X,$$ $$\Omega(X_{\geq 0})=\Omega(Y_{>0}),$$

and each face positive geometry $(C,C_{\geq 0})$ of $(X,X_{\geq 0})$ is also a compactification of a cluster variety and its positive part. This would extend the positive-geometry structure seen in positroid and cluster-variety examples, but the existence of such compactifications with the stated face structure is not established in general.