The graph C*-algebra classification conjecture by filtered ordered K-theory

Let $E$ be a graph and let $C^*(E)$ be its graph $C^*$-algebra. Suppose that $C^*(E)$ has finitely many ideals, and write ${\mathsf{FK}^+_{\operatorname{Prim}(C^*(E))}\left(C^*(E)\right)}$ for its filtered, ordered $K$-theory. Graph C-algebra classification conjecture.* Graph $C^*$-algebras $C^*(E)$ with finitely many ideals are classified up to stable isomorphism by their filtered, ordered $K$-theory ${\mathsf{FK}^+_{\operatorname{Prim}(C^*(E))}\left(C^*(E)\right)}$. The conjecture is a proposed classification of graph $C^*$-algebras by $K$-theoretical invariants. It remains open, and some results suggest that an additional condition of finitely generated $K$-theory may be necessary.