The graded classification conjecture for finite Leavitt path algebras

Let $E$ and $F$ be finite graphs, let $L(E)$ and $L(F)$ be their Leavitt path algebras over a field $\mathsf{k}$, and let $K_0^{\operatorname{gr}}$ be the graded Grothendieck group, viewed as an ordered $\mathbb{Z}[x,x^{-1}]$-module with distinguished identity class. An order-preserving module isomorphism is required to preserve this distinguished class.

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

$$\phi:K_0^{\operatorname{gr}}(L(E))\overset{\sim}{\longrightarrow}K_0^{\operatorname{gr}}(L(F))$$

with $\phi([L(E)])=[L(F)]$ if and only if there exists a unital $\mathbb{Z}$-graded $\mathsf{k}$-isomorphism $\psi:L(E)\rightarrow L(F)$ such that $K_0^{\operatorname{gr}}(\psi)=\phi$.

This conjecture extends the classification theorem known for polycephaly graphs to general finite graphs; the source also notes positive results for amplified graphs and classification up to twisting in a related setting.