Hazrat–Roozbeh's graph-algebra classification conjecture

Let $E$ and $F$ be finite graphs, let $k$ be a field, let $C^*(E)$ and $C^*(F)$ denote their graph $C^*$-algebras, let $L_k(E)$ and $L_k(F)$ denote their Leavitt path algebras, and let \H_0^{\operatorname{gr}}(\mathcal{G}_E) and $H_0^{\operatorname{gr}}(\mathcal{G}_F)$ be the zeroth graded homology groups of the associated étale graph groupoids. Hazrat–Roozbeh's conjecture. The following are equivalent: (1) there is a gauge-preserving isomorphism $C^*(E)\to C^*(F)$; (2) there is a graded ring isomorphism $L_k(E)\to L_k(F)$; and (3) there is an order-preserving $\mathbb Z[x,x^{-1}]$-module isomorphism

$$H_0^{\operatorname{gr}}(\mathcal{G}_E)\longrightarrow H_0^{\operatorname{gr}}(\mathcal{G}_F)$$

that preserves the order unit, sending $[1_{\mathcal{G}_E^{(0)}}]$ to $[1_{\mathcal{G}_F^{(0)}}]$. This conjecture proposes a common classification invariant for finite graph $C^*$-algebras and Leavitt path algebras; its resolution is not specified in the source.