The graded homomorphism and isomorphism conjecture for Leavitt path algebras

Let $E$ and $F$ be finite graphs, let $\mathsf{k}$ be a field, and write $L_{\mathsf{k}}(E)$ and $L_{\mathsf{k}}(F)$ for their Leavitt path algebras. Their graded Grothendieck groups $K_0^{\operatorname{gr}}(L_{\mathsf{k}}(E))$ and $K_0^{\operatorname{gr}}(L_{\mathsf{k}}(F))$ are ordered $\mathbb Z[x,x^{-1}]$-modules, with distinguished classes of the algebras. The graded homomorphism and isomorphism conjecture. For any order preserving $\mathbb Z[x,x^{-1}]$-module homomorphism

$$\phi:K_0^{\operatorname{gr}}(L_{\mathsf{k}}(E))\longrightarrow K_0^{\operatorname{gr}}(L_{\mathsf{k}}(F))$$

with $\phi([L_{\mathsf{k}}(E)])=[L_{\mathsf{k}}(F)]$, there exists a unital $\mathbb Z$-graded $\mathsf{k}$-homomorphism $\psi:L_{\mathsf{k}}(E)\to L_{\mathsf{k}}(F)$ such that $K_0^{\operatorname{gr}}(\psi)=\phi$. Moreover, if $\phi$ is an isomorphism, then there exists a unital $\mathbb Z$-graded $\mathsf{k}$-isomorphism $\psi:L_{\mathsf{k}}(E)\to L_{\mathsf{k}}(F)$ such that $K_0^{\operatorname{gr}}(\psi)=\phi$. The first part has been answered affirmatively by G. Arnone and L. Vas, so this candidate is solved; the source evidence does not establish the status of the second part independently.