Weak closedness of indecomposable traces for AF-algebras of distributive lattices

Let $\Gamma$ be a countable distributive lattice, realized as the lattice of finite ideals of a partially ordered set, and let $A(\Gamma)$ be the $AF$-algebra associated with the Hasse--Bratteli diagram of $\Gamma$. Let $\Bbb Y^n$ denote the Hasse diagram of the lattice of finite ideals of $\Bbb Z_+^n$, and let $A(\Bbb Y^n)$ be its associated $AF$-algebra. Weak-closedness conjecture. The list of indecomposable traces of $A(\Gamma)$ is weakly closed; consequently, $A(\Gamma)$ is completely smooth. In particular, the same holds for $A(\Bbb Y^n)$. The conjecture proposes a generalization of the measure-theoretic approach to filtrations from homogeneous partitions to semi-homogeneous partitions, with the goal of obtaining information about invariant measures and traces of $AF$-algebras. The source does not provide evidence resolving this claim.