The ring-versus-algebra graded isomorphism conjecture for Leavitt path algebras

Let $E$ and $F$ be row-finite graphs over a field $K$. Write $\operatorname{\mathcal L}_K(E)$ and $\operatorname{\mathcal L}_K(F)$ for their Leavitt path algebras, and let $\cong_{\operatorname{gr}}$ denote a graded isomorphism. Ring-versus-algebra graded isomorphism conjecture.

$$\operatorname{\mathcal L}_K(E)\cong_{\operatorname{gr}}\operatorname{\mathcal L}_K(F)\text{ as rings}$$

if and only if

$$\operatorname{\mathcal L}_K(E)\cong_{\operatorname{gr}}\operatorname{\mathcal L}_K(F)\text{ as $K$-algebras}.$$

The claim asks whether every graded ring isomorphism between these Leavitt path algebras is automatically a graded $K$-algebra isomorphism. In the source it is posed for the class of polycephaly graphs; no resolution beyond the stated results is supplied.