Hazrat's full-faithfulness conjecture for the graded Grothendieck-group functor

Let $\ell$ be a field. Consider the category of unital Leavitt path $\ell$-algebras with graded homomorphisms modulo inner automorphisms, and the category of preordered abelian groups with order unit. Hazrat's full-faithfulness conjecture. The graded Grothendieck group is a fully faithful functor from the first category to the second. The conjecture is refuted in full generality: the supplied status evidence states that its faithfulness part fails, although the paper establishes the fullness part.