Hazrat's strong classification conjecture for unital Leavitt path algebras

A unital Leavitt path algebra is a Leavitt path algebra arising from a finite graph, equipped with its natural grading. Let the source category have these algebras as objects and graded homomorphisms modulo inner automorphisms as morphisms. Let the target category consist of pre-ordered abelian groups with order-unit, and let $K_0^{\operatorname{gr}}$ denote the graded Grothendieck-group functor.

Hazrat's strong classification conjecture. The functor $K_0^{\operatorname{gr}}$ is fully faithful from the category of unital Leavitt path algebras with graded homomorphisms modulo inner automorphisms to the category of pre-ordered abelian groups with order-unit.

This is a stronger functorial classification claim than classification by the invariant on objects alone, because it also requires all morphisms in the target to arise uniquely up to the stated equivalence. The source attributes it to Hazrat as Conjecture 3 and gives no evidence of resolution.