The graded homology classification conjecture for finite graphs

Let $E$ and $F$ be finite graphs, and let $R$ be a field. Write $\mathcal{G}_E$ and $\mathcal{G}_F$ for their graph groupoids, and let $H_0^{\operatorname{gr}}$ denote graded groupoid homology. The distinguished classes $[1_{\mathcal{G}_E^0}]$ and $[1_{\mathcal{G}_F^0}]$ come from the unit spaces. Graded homology classification conjecture. The following are equivalent: there is a gauge-preserving isomorphism $\phi:C^*(E)\rightarrow C^*(F)$; there is a graded ring isomorphism $\phi:L_R(E)\rightarrow L_R(F)$; and there is an order-preserving $\mathbb{Z}[x,x^{-1}]$-module isomorphism

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

that sends $[1_{\mathcal{G}_E^0}]$ to $[1_{\mathcal{G}_F^0}]$. This conjecture seeks a common classification invariant for graph $C^*$-algebras, Leavitt path algebras, and their graded groupoid homology; the source indicates that it is motivated by previous work but does not state a resolution.