The fully faithful graded Grothendieck group conjecture for Leavitt path algebras

Let $K_0^{\operatorname{gr}}$ denote the graded Grothendieck group functor. Consider the category whose objects are Leavitt path algebras and whose morphisms are graded homomorphisms modulo inner automorphisms; its target is the category of pre-ordered abelian groups with order-units. Fully faithful graded Grothendieck group conjecture. The functor

$$K_0^{\operatorname{gr}}$$

is fully faithful from the category of Leavitt path algebras with graded homomorphisms modulo inner automorphisms to the category of pre-ordered abelian groups with order-units.

The conjecture would extend the paper's full-faithfulness result from acyclic Leavitt path algebras to all Leavitt path algebras, making the graded Grothendieck group a complete morphism-level invariant in this categorical setting. The general assertion remains open in the source.