The Algebraic Kirchberg-Phillips Question for Leavitt path algebras

The Algebraic Kirchberg-Phillips Question for Leavitt path algebras asks whether unital K-theory is a complete isomorphism invariant for unital, simple, purely infinite Leavitt path algebras over finite graphs. Most work on this problem has focused on determining whether (up to isomorphism) there is a unique unital, simple, Leavitt path algebra with trivial K-theory (often reformulated as the question of whether the Leavitt path algebras L₂ and L2₋ are isomorphic). However, it is unknown whether a positive answer to this special case implies a positive answer to the Algebraic Kirchberg-Phillips Question. In this note, we pose a different question that asks whether two particular non-simple Leavitt path algebras Lₖ(F_*) and Lₖ(F**) are isomorphic, and we prove that a positive answer to this question implies a positive answer to the Algebraic Kirchberg-Phillips Question.