The graph extension conjecture for unital graph $C^*$-algebras

Let $C^*(E_1)$ and $C^*(E_3)$ be unital graph $C^*$-algebras with finitely many ideals, and consider the unital extension

$$0\longrightarrow C^*(E_1)\otimes \mathbb{K}\longrightarrow \mathfrak{X}\longrightarrow C^*(E_3)\longrightarrow 0.$$

Graph extension conjecture. The following are equivalent:

1. $\mathfrak{X}$ is a graph $C^*$-algebra.
2. $\mathfrak{X}$ has real rank zero.
3. The map $\partial_0:K_0(C^*(E_3))\to K_1(C^*(E_1))$ vanishes.

This conjecture seeks a complete characterization of when such unital extensions belong to the class of graph $C^*$-algebras. The status evidence indicates that Bentmann provided a complete solution in the Cuntz–Krieger case, while this broader formulation is recorded as a conjecture.