The complex Leavitt path algebra star-isomorphism conjecture

Let $E$ and $F$ be directed graphs, and let $L_{\a0\mathbb{C}}(E)$ and $L_{\a0\mathbb{C}}(F)$ denote their Leavitt path algebras over the complex numbers, with the complex-conjugate involution. Complex star-isomorphism conjecture. If

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

as algebras, then

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

as $^\ast$-algebras. This is one of the isomorphism conjectures posed by Abrams and Tomforde; the supplied text does not state whether it has been resolved.