The fullness conjecture for graded Grothendieck groups of Leavitt path algebras

Let $E$ and $F$ be 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 functor to pre-ordered $\mathbb{Z}[x,x^{-1}]$-modules with order unit. Let $\phi:K_0^{\operatorname{gr}}(L(E))\rightarrow K_0^{\operatorname{gr}}(L(F))$ be an order-preserving module homomorphism preserving the order unit.

Hazrat's fullness conjecture. There exists a unital $\mathbb{Z}$-graded $\mathsf{k}$-homomorphism $\psi:L(E)\rightarrow L(F)$ such that

$$K_0^{\operatorname{gr}}(\psi)=\phi.$$

The source states that this conjecture was proved independently by Arnone and Vaš; therefore the conjecture is no longer open.