The equivariant classification conjecture for graph $C^*$-algebras

Let $E$ and $F$ be finite graphs, and let $C^*(E)$ and $C^*(F)$ be their graph $C^*$-algebras with gauge circle actions. Let $K^{\mathbb T}_0$ denote equivariant $K$-theory, viewed as an ordered $\mathbb Z[x,x^{-1}]$-module.

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

$$\phi:K^{\mathbb T}_0(C^*(E))\to K^{\mathbb T}_0(C^*(F)),$$

with $\phi([C^*(E)])=[C^*(F)]$, if and only if $C^*(E)$ and $C^*(F)$ are isomorphic by an isomorphism respecting the gauge action.

This is the analytic counterpart of the graded classification conjecture for Leavitt path algebras, using the identification between graded and equivariant $K$-theory. The source gives no resolution status.