The equivariant K-theory classification conjecture for graph C*-algebras

Let $E$ and $F$ be finite graphs. Write $C^*(E)$ and $C^*(F)$ for their graph $C^*$-algebras, let $\gamma_E$ and $\gamma_F$ denote the gauge circle actions, and let $K_0^{\mathbb{T}}$ denote $\mathbb{T}$-equivariant $K$-theory. An order-preserving $\mathbb{Z}[x,x^{-1}]$-module isomorphism is required to preserve the distinguished class of the unit.

Equivariant K-theory classification conjecture. There is an order-preserving $\mathbb{Z}[x,x^{-1}]$-module isomorphism

$$\phi:K_0^{\mathbb{T}}(C^*(E))\overset{\sim}{\longrightarrow}K_0^{\mathbb{T}}(C^*(F))$$

with $\phi([C^*(E)])=[C^*(F)]$ if and only if there is a $\mathbb{T}$-equivariant $*$-isomorphism $C^*(E)\overset{\sim}{\longrightarrow}C^*(F)$.

This is the analytic counterpart of the graded classification conjecture, using the correspondence between graded $K$-theory of Leavitt path algebras and equivariant $K$-theory of graph $C^*$-algebras. The source presents it as conjectural.