The graded Grothendieck-group classification conjecture for finite Leavitt path algebras

Let $E_1$ and $E_2$ be finite graphs and $F$ a field. Write $M^{\operatorname{gr}}_{E_i}$ for the graded graph monoid and $K_0^{\operatorname{gr}}(L_F(E_i))$ for the graded Grothendieck group of the Leavitt path algebra, with its order and $\mathbb Z[x,x^{-1}]$-module structure. The graded Grothendieck-group classification conjecture. The following are equivalent: there is a $\mathbb Z$-module isomorphism $\phi:M^{\operatorname{gr}}_{E_1}\rightarrow M^{\operatorname{gr}}_{E_2}$ such that $\phi\bigl(\sum_{v\in E_1^0}v\bigr)=\sum_{v\in E_2^0}v$; there is an order-preserving $\mathbb Z[x,x^{-1}]$-module isomorphism

$$K_0^{\operatorname{gr}}(L_F(E_1))\longrightarrow K_0^{\operatorname{gr}}(L_F(E_2))$$

sending $[L_F(E_1)]$ to $[L_F(E_2)]$; and there is a graded ring isomorphism $\varphi:L_F(E_1)\rightarrow L_F(E_2)$. The conjecture proposes that the graded Grothendieck group, together with its ordering, module structure, and distinguished class, is a complete invariant for finite Leavitt path algebras. The equivalence with graded ring isomorphism is the classification assertion; the supplied text does not state whether it has been proved or disproved.