The Abrams–Tomforde conjecture for Leavitt path algebras and graph $C^*$-algebras

Let $E$ and $F$ be graphs and let $\mathsf{k}$ be a field. Consider the Leavitt path algebras $L_\mathsf{k}(E)$ and $L_\mathsf{k}(F)$, and the graph $C^*$-algebras $C^*(E)$ and $C^*(F)$. Abrams–Tomforde conjecture. The following statements are equivalent:

1. $L_\mathsf{k}(E)$ and $L_\mathsf{k}(F)$ are isomorphic as rings.
2. $C^*(E)$ and $C^*(F)$ are isomorphic as $C^*$-algebras.

This conjecture, posed in 2011, seeks to characterize when the algebraic and analytic graph-algebra constructions have the same isomorphism classification. The parser supplies no resolution evidence, so it is recorded as open.