The complex Leavitt path algebra to graph C-star algebra conjecture

Let $E$ and $F$ be directed graphs, let $L_{\mathbb{C}}(E)$ and $L_{\mathbb{C}}(F)$ be their Leavitt path algebras, and let $C^*(E)$ and $C^*(F)$ be the associated graph $C^*$-algebras. Complex graph $C^*$-algebra conjecture. If

$$L_{\mathbb{C}}(E)\cong L_{\mathbb{C}}(F)$$

as algebras, then

$$C^*(E)\cong C^*(F)$$

as $^\ast$-algebras. The source notes that this follows from C1 if the latter holds, while only the implication from a $^\ast$-isomorphism of Leavitt path algebras is known; the conjecture itself is not resolved in the supplied text.